literally
#help-36
steady locust
the first is the indefinite integral, its the anti-derivative. the second is the definite integral, it computes to a number
carmine plaza
seems vague
steady locust
the anti-derivative is the inverse of the derivative, its a function. the definite integral is the area under the curve
$\int{f(x)}dx$ = anti-derivative of f(x), $\int_a^b{f(x)}dx$ = definite integral of f(x) from a to b
carmine plaza
i mean the derivative represents a tangent line
the integral represents the area under the curve
soft zealotBOT
c2b7
carmine plaza
if its indefinite it just represents the entire area under the curve at any given point
steady locust
the integral represents the area under the curve
the "area under the curve" is the one with a and b
steady locust
if its indefinite it just represents the entire area under the curve at any give...
yeah, it would be a function
steady locust
not really, because it depends on where you start for the area under the curve, thats why the indefinite integral has + C
have you learned doing indefinite integral / antiderivative?
supple jolt
The indefinite integral just gives you a function that you use to compute the area
carmine plaza
right
steady locust
yeah, but you need to compute over an interval, not just at a value
supple jolt
I mean you can go from a to a
carmine plaza
hm
supple jolt
But yes
I suggest watching a youtube video about this
Its kinda hsrd to explain over text
steady locust
I mean you can go from a to a
which would still just be an interval with 0 measure
carmine plaza
i guess its not rly something that matters
supple jolt
Visuals help
carmine plaza
i dont need to necessarily know this
steady locust
i dont need to necessarily know this
what have you gone over so far?
carmine plaza
so if an integral doesnt have bounds it cant be improper
supple jolt
Do you have like a specific question or just want to know what integrals are in general
steady locust
so if an integral doesnt have bounds it cant be improper
yeah, improper is only for definite integrals
steady locust
Do you have like a specific question or just want to know what integrals are in ...
bro its fine to ask a general question
carmine plaza
i just didnt understand that difference
spring haven
Indefinite integrals aren’t real
steady locust
Indefinite integrals aren’t real
shut up
carmine plaza
i feel like thats true XD
spring haven
is he wrong idk
I was joking lol but you don’t really see those after a point
carmine plaza
like thats why i was confused i always assumed it was shorthand for an improper integral between -inf to inf
that indefinite was
steady locust
I was joking lol but you don’t really see those after a point
yeah, its not helpful, so dont say it
you just confuse people
carmine plaza
I was joking lol but you don’t really see those after a point
just didnt make sense for them to not have a bound
steady locust
like thats why i was confused i always assumed it was shorthand for an improper ...
no, indefinite vs definite are completely different things
they are related by the fundamental theorem of calculus
supple jolt
If they are bounded the reuslt is a number
Of they are unbounded the result is a function
Think of it this way
For now
carmine plaza
alr everything else makes sene
ill open something else if i have another question
.close
final saddleBOT
supple jolt
Right
Just watch a youtube video
If you want
Its easier to understand
Your choice
carmine plaza
i understand i mean that wasnt the issue
supple jolt
All right
final saddleBOT
Available help channel!
last forum
short question for example if im trying to solve
-3 -1/5 am i supposed to multiply or not
jagged flare
$-3-\frac{1}{5}$
soft zealotBOT
ihave<skissue>
jagged flare
this?
last forum
yes
i asked chatgpt and it kept going back and fourth not knowing if the right way was to not multiply or to multiply
jagged flare
thats a subtraction, -3 then subtract a fifth
tired walrus
i asked chatgpt and it kept going back and fourth not knowing if the right way w...
!noai
final saddleBOT
i asked chatgpt and it kept going back and fourth not knowing if the right way w...
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
tired walrus
anyway -3 - 1/5 is quite clearly a subtraction of 1/5 from -3
tired walrus
don't be surprised if math turns out to be 17 years of great pain and suffering.
last forum
anyway -3 - 1/5 is quite clearly a subtraction of 1/5 from -3
thx
full question btw
tired walrus
its pain and suffering now lol
and you have 17 more years of it in store if you don't stop using GPT to do math
tired walrus
ok so you do in fact subtract 1/5 off both sides here and you end up with x/3 = -16/5
tired walrus
ok, so do it with steps.
tired walrus
i am telling you very slightly in advance what will happen
you can check yourself against me -- if you arrive at the same answer as me, it means you made an even number of mistakes
last forum
you can check yourself against me -- if you arrive at the same answer as me, it ...
ik i got something wrong
ignore messy drawing i hate mouse
tired walrus
the -3.2 bit is ok
then you divided by 1/3 on the left (also good), but... subtracted 3 on the right?
you were meant to multiply by 3 on both
the golden rule is you do the SAME THING on both sides
last forum
i think i subtracted because it was positive and i moved it over so its negative
tired walrus
i think i subtracted because it was positive and i moved it over so its negative
this is why thinking in terms of "moving" is bad 💔
last forum
its just so i can visualize it better
moving over making it negative yk
thats why im crossing out
chatgpt mightve corrupted me
if a fraction has a - in between you can still multiply?
😵💫
actually i get it
i was dividing it not subtracting it so theres no need for it to be negative
smh
tiny kraken
Let A be a set containing n elements .A partition of a set is a set of k non-empty subsets such that
$$A = A_1 \cup A_2 \cup A_3 \cup \dots \cup A_i , ( A_i \cap A_j = \o , \forall i,j = 1;2;\dots ;k )$$
For example: $B={a,b}$ has 2 partitions ${{a},{b}}$ and ${{a,b}}$ \
How many partitions of a set with 4 elements $C={a,b,c,d}$ are there?
final saddleBOT
Let A be a set containing n elements .A partition of a set is a set of k non-emp...
soft zealotBOT
Fionna The Unemployed
rugged merlin
What have you tried
silver dew
There is a recursion formula for the Bell number. You can use that
tiny kraken
Okay so, I was able to do this problem using recurrence but I couldn't find a closed form or at least non-recursively relationship
tiny kraken
There is a recursion formula for the Bell number. You can use that
eh?
Is it like
$A_k = A_{k-1} + \binom{n}{1} A_{k_2} + \dots$
soft zealotBOT
Fionna The Unemployed
hasty mist
The Bell number is the sum of all Stirling numbers of the second kind
silver dew
You can easily deduce it, by choosing some k elements as the last set, and n-k elements have their own division:
tiny kraken
**Fionna The Unemployed**
I tried to find closed form of this recurrence closed form
silver dew
$B_{n+1}=\sum_{k=0}^{n}\binom{n}{k}B_{k}$
soft zealotBOT
Cogwheels of the mind
silver dew
Yeah
tiny kraken
The Bell number is the sum of all Stirling numbers of the second kind
Okay I also have no idea what Stirling numbers are
silver dew
Sorry, I am going to reformulate the thought again:
hasty mist
Okay I also have no idea what Stirling numbers are
S(n,k) counts the number of partitions of n elements into exactly k non-empty subsets.
silver dew
Dividing the {1,2,…,n+1} is the same as choose n-k elements from {1,2,…,n}, along with n+1 to make a subset, and B_k many ways to divide the remaining of {1,2,…,n}. For k=0,1,…,n
hasty mist
[ rb{ B_n = sum_{k=0}^n S(n,k) }]
soft zealotBOT
クーリー
tiny kraken
What I did was, choose an element m from the set A then doing cases work, if m standalone then it has the amount of partitions of a set with 3 elements ways and so on
tiny kraken
**Cogwheels of the mind**
I did arrived at this formula
silver dew
Dividing the {1,2,…,n+1} is the same as choose n-k elements from {1,2,…,n}, alon...
So essentially it is dividing it to cases according to where n+1 locates
silver dew
**Cogwheels of the mind**
^
tiny kraken
I know
silver dew
B_1=1, good to go.
tiny kraken
I mean, like how Fibonacci numbers also have a closed form
Does bell numebers have one?
silver dew
I don’t think so
hasty mist
Yeah,
Dobinski
It’s not very practical though
[rb{B_n = frac1e sum_{k=0}^{oo} frac{k^n}{k!} }]
soft zealotBOT
クーリー
hasty mist
It follows from the fact that a Poisson(1) random number of labels produces partitions with expected count exactly equal to the Bell number.
hasty mist
Yes
tiny kraken
Okay I'm still a highschooler
silver dew
The last thing he mentioned probably is. Beyond that no, all elementary
hasty mist
You seem unemployed
tiny kraken
You seem unemployed
You're absolutely right
hasty mist
I knew it
silver dew
But if you are interested in combinatorics in general you can always read enumerative combinatorics by Stanley. It’s beautiful
tiny kraken
I'll give it a try, the exam I'm going to take are full of combinatorics problems
silver dew
If you have an exam ahead then don’t mind. I meant a hobby kind of thing in free time
tiny kraken
If you have an exam ahead then don’t mind. I meant a hobby kind of thing in free...
the only thing I love is watching cartoon but it's okay I also love to torture myself sometimes
Thanks y'all
.close
final saddleBOT
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spring abyss
about multivar differentiation chain rule, does wuu = wxx * (xu)^2 ?
since d^2w/dx^2 needs to * dx twice and / du twice if treating them as numbers on simple arithmetic
final saddleBOT
about multivar differentiation chain rule, does wuu = wxx * (xu)^2 ?
since d^2w...
main mirage
about multivar differentiation chain rule, does wuu = wxx * (xu)^2 ?
since d^2w...
Mmmm well,
When in doubt - you can always take derivatives manually!
main mirage
about multivar differentiation chain rule, does wuu = wxx * (xu)^2 ?
since d^2w...
By the way, what is w a function of?
Is w a function of x, and x depends on u?
spring abyss
lets just say, wu exists, wx exists, and xu exists
rugged merlin
about multivar differentiation chain rule, does wuu = wxx * (xu)^2 ?
since d^2w...
This is not correct btw
main mirage
lets just say, wu exists, wx exists, and xu exists
Yeah I mean like what is w a function of
Like
w = w(x)?
spring abyss
idk
main mirage
Okay so w is a function of x and t, since its the wave equation.
main mirage
idk
So here's the deal.
We want to write w, a function of x and t: w(x, t) into w a function of u and v w(u, v), and namely its derivatives as well. (The reason being is because the question wants us to substitute u and v in place of x and t)
What does the multivariable chain rule say about taking the derivative? I.e
If
w = w(x, t)
How do we figure out what dw/du is?
spring abyss
wu = wx * xu
main mirage
wu = wx * xu
Mmmm unfortunately not :(
The problem is that u depends on both x and t, so this equation is actually missing something.
main mirage
Good! Because u depends on x and t, you are effectively doing the chain rule for each variable.
spring abyss
then another question, how to expand this concept onto 2nd d?
wxx = wuu . uxx +...
or wxx = wuu . (ux)^2 +... ?
wait
main mirage
wu = wx . xu + wt . tu
It will get very annoying if you derive the second derivative expression. The reason being is because wx * xu is yet another multivariate function.
spring abyss
wxx = d/dx(wx) where i gotta expand both d/dx and wx?
main mirage
That means you have to apply the multivariable chain rule to wx * xu and wt * tu separately, which is not as easy algwbraically as it might seem.
main mirage
wxx = d/dx(wx) where i gotta expand both d/dx and wx?
Wdym
spring abyss
wx has its own expression, then d/dx becomes d/du . ux + d/dt . tx
spring abyss
wuh
main mirage
As a hint, the strategy for this question only requires you to find wuu and ||wtt||.
If you can find those two, you can solve the question :)
Well actually
You dont even need to find those two
The question tells you exactly which derivative you need to find.
main mirage
I’ll look into it later then
Sure 👍
spring abyss
Thx
main mirage
The question tells you exactly which derivative you need to find.
This is not so much a trick question by the way - if you look at the question once more and see what it wants you to find/prove
What derivative do you think you need to compute?
(Anyway take a break if you need it :)
final saddleBOT
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Available help channel!
languid bay
how do you solve this? I made it into a quadratic equation, solved for a in 2 diff values like a = -3 and a =1 , and I substituted the a values for both the equation, I get 9 and 809/27. however the answer is 71/27 is there anywhere I'm going wrong
final saddleBOT
how do you solve this? I made it into a quadratic equation, solved for a in 2 di...
smoky bramble
do you know the factorization of a^3-b^3?
if yes, you should be able to factorize a^3-1/a^3
languid bay
not very sure, but like how do you apply it?
smoky bramble
3a-3/a=3(a-1/a)
languid bay
and you solve for a? or do we like directly substitute it in?
smoky bramble
and you solve for a? or do we like directly substitute it in?
i don't understand what you are asking for
main mirage
and you solve for a? or do we like directly substitute it in?
The short answer is no, you don't need to solve for a :)
languid bay
can we not substitute a-1/a into the second equation?
main mirage
As d1 implicitly mentioned, you are basically just doing some neat substitution tricks.
smoky bramble
can we not substitute a-1/a into the second equation?
if you can express a-1/a in terms of a³-1/a³
why not?
languid bay
so in that case I can take the value to be -4/3 and solve?
languid bay
yess
final saddleBOT
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Available help channel!
tranquil charm
how to calculate Z-transform of n/n! ?
final saddleBOT
how to calculate Z-transform of n/n! ?
tranquil charm
i know that Z-ransform of 1/n! is e^{1/z}
trail mango
what’s another way to write n/n! that looks similar to 1/n!
tranquil charm
what’s another way to write n/n! that looks similar to 1/n!
1/(n-1)!
trail mango
yep
tranquil charm
alright but what now?
trail mango
let me see…
trail mango
yea i think it should just shift nicely
tranquil charm
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final saddleBOT
final saddleBOT
Available help channel!
tranquil charm
how to calculate $\mathcal{Z}$-transform of $\ln\left(\frac{z-1}{z}\right)$?
soft zealotBOT
Slowaq
tranquil charm
.close
final saddleBOT
final saddleBOT
Available help channel!
scenic nest
I am trying for an optimized Gaussian elimination, should I only stop when var=number and at the end? I have so far implemented ranking where the first variable is, is like var=number and the first variable in the diagonal being the thing to subtract from the next rows enough or should I full sort?
final saddleBOT
I am trying for an optimized Gaussian elimination, should I only stop when var=n...
scenic nest
Note this is parallel on the GPU. X E.
I also have implemented the stop. X E.
Is this better with or without sorting the rows?
Should I just continue implementing and see if the not sort can work?
vital crag
if you don't need to perform an operation, why would you need to
scenic nest
Oh, is it possible to just not even use a maximum?
scenic nest
Yeah, at worst O(n^3), is there something better I should consider?
Just as optimized as I can make it. X E.
Just fastest I can make it with minimum extra storage. X E.
Thanks for talking. X E.
vital crag
it really depends on the use case. see
https://en.wikipedia.org/wiki/Gaussian_elimination#Computational_efficiency
desert mantle
there are highly highly optimized versions of gaussian elimination available
why would you want to code it yourself
desert mantle
nothing you can produce will come close to what others have done
scenic nest
why would you want to code it yourself
Are they in Slang shaders?
scenic nest
there are highly highly optimized versions of gaussian elimination available
Okay, can you name one to study?
Thanks for talking. X E.
desert mantle
any useful language should have a linear algebra library available
which has gaussian elimination already implemented
vital crag
any useful language should have a linear algebra library available
rip ruby
scenic nest
Those... open source? How easy are those to glean from?
vital crag
start with a book specifically about matrix computations. maybe
https://books.google.com/books/about/Matrix_Computations.html?id=X5YfsuCWpxMC
scenic nest
I already have a basic understanding of matrices. X E.
vital crag
sciruby used to be good, but it hasn't been updated in 4 years
https://github.com/SciRuby/sciruby.com
scenic nest
I am finding a lot of CPU gaussian elimination, any GPU?
vital crag
tepid ridge
google is ✨
https://cs.ucsb.edu/sites/default/files/documents/2008-15.pdf
do you understand anything or relying on google only ?
scenic nest
Come on internet, GPU parallel non-square. X E.
drowsy epoch
do you understand anything or relying on google only ?
i prompt everything, google is for uncs
tepid ridge
i prompt everything, google is for uncs
they are still stuck on google
drowsy epoch
o oki
vital crag
do you understand anything or relying on google only ?
do you not know how to use the internet?
your mom must have installed discord for you
tepid ridge
do you not know how to use the internet?
doesn’t seem like you are explaining you are letting them rely on google only
do not claim ticket if you do not know how to solve
vital crag
ask your mom to explain to you how discord works next
vital crag
do not claim ticket if you do not know how to solve
if you can answer questions, do it. no one's stopping you
scenic nest
https://github.com/ShantaoL/PRBP, is this any good?
tepid ridge
if you can answer questions, do it. no one's stopping you
Maybe not now
tepid ridge
I will do that once I arrive home
vital crag
Maybe not now
then stop flooding help channels with useless sentences
tepid ridge
then stop flooding help channels with useless sentences
I had to stop by and see what you were doing because I don’t think the person understands what google stuff your sending
next are you gonna use ai?
scenic nest
https://github.com/ShantaoL/PRBP, is this any good?
Well, that was the only open source one I found that was good sort of and it is ten years old. X E.
scenic nest
next are you gonna use ai?
Not me. X E.
vital crag
I had to stop by and see what you were doing because I don’t think the person un...
<@&268886789983436800>
tepid ridge
<@&268886789983436800>
??
scenic nest
Well, is there another place to look other than GitHub?
desert mantle
google...?
you can for example also find https://docs.nvidia.com/cuda/cusolver/index.html
how about you actually look
but serious question, do you actually need this whole completely optimized shit. or are you just gonna use it to solve 10x10 systems
scenic nest
google...?
Yeah, not much. X E.
vital crag
Yeah, not much. X E.
you need to spend more than 1 minute googling
scenic nest
I spent most of today and also exhausted GitHub. X E.
Codeberg turns up no entries. X E.
Maybe I am just bad at searching?
vital crag
you need specific search terms
scenic nest
And it can't be like use our algorithm, it must be like look at ours, feel free to copy. X E.
Nothing is in Slang shaders now so if nothing else I will at least translate. X E.
But the no pivot solver gives me hope. X E.
faint edge
I had to stop by and see what you were doing because I don’t think the person un...
Please keep whatever personal beef you have with another users attempt to help out of the help channels. It's just going to get in the way of folks getting help.
scenic nest
Anyway, back to the original question, do I need sort or just like put the frontmost variable equation up top to zero out the rest?
I would think any extra work will degrade performance. X E.
Should I just try myself and later ask why it does or does not work?
vital crag
you can for example also find <https://docs.nvidia.com/cuda/cusolver/index.html>
Yes. Or just use an existing library
vital crag
but serious question, do you actually need this whole completely optimized shit....
And answer this first
scenic nest
Yes. Or just use an existing library
Sorry, can't use an existing library unless it is already for all the GPU APIs. X E.
vital crag
you can for example also find <https://docs.nvidia.com/cuda/cusolver/index.html>
Did you not read this at all
scenic nest
but serious question, do you actually need this whole completely optimized shit....
Yes, I might do it on 20000 by 20003 matrices. X E.
scenic nest
Did you not read this at all
I did, historically Nvidia is hard to find the actual algorithms from. X E.
I read some. X E.
molten ginkgo
Do u guys know someone's explaining How to solve a linear programming problem using the graphical method?
scenic nest
Do u guys know someone's explaining How to solve a linear programming problem us...
Not for here?
quasi rapids
Do u guys know someone's explaining How to solve a linear programming problem us...
Not your channel mate
molten ginkgo
Not your channel mate
Isn't that a mathematical server ?
molten ginkgo
Not for here?
Wdym ?
quasi rapids
This help channel is occupied
If you have your own question, make your own help channel
molten ginkgo
This help channel is occupied
Oh okay I'm sorry
scenic nest
Well, going through the given link, it does not actually have algorithms, just a use our system ad. X E.
If you can find the repo form great. X E.
And something slow is better than nothing, without an example code, it is nothing. X E.
What? Second to last channel down?
Anyway, either provide an example I can copy or answer the pinned question please?
And all numbers are real. X E.
This is also 4d+. X E.
What does finding a no pivot one mean?
Hi, welcome. X E.
What is a pivot in Gaussian elimination?
I know, I should just look it up. X E.
Oh, the top row that is setting others zero is the pivot. X E.
That no pivot is just wrong. X E.
AI says I can, but blah, AI. X E.
I think just place any first place should work, and don't interfere with solutions should work. X E.
I would almost thank people here if this had come from anyone else. X E.
GPU gaussian elimination is scarce. X E.
I am modeling it on the CPU in Rust. X E.
At least on CPU I know like line. X E.
Any other Gaussian elimination tips?
I think this can work. X E.
Okay, I think this goes in stages, like first set ranks and go for maximum, search for singles, if found then substitute, next set matrix with below top one only, a thread per each element subtracting except the first which stays the same and is needed, then check for single variables, then set that to zero unless single, if single found, use it. Could this be more optimized? X E.
After the set up matrix repeats. X E.
Maybe I should have a single vector like thing of found variables and zero out singles. X E.
So rank->move up one->eliminate one->check singles and place in vector->zero out not singles->start. X E.
Rank can be done parallel per equation with end at first, move up one can be done with a single swap single thread, eliminate one big number of threads, check singles per equation, maybe a next step to move that variable to equals side per equation between, then zero it out in the equations, then start. X E.
Is this efficient or trash 🗑️?
Also the startup is like rank, move up one input, set up matrix not including that index, check and substitute in singles, start. X E.
Also, I should probably have like if absolute value less than some epsilon then zero. X E.
Oh, and also, may as well stop the single variable checker at second variable and if one contains zero variables and last element like after equals is not zero then atomically set a variable, then at the end of the iteration a checker moves it to non-volatile memory, every thread checks that before continuing on the GPU. X E.
I could probably have just addition and if integer=0. X E.
Should I just implement and stop bothering you?
Simplicity wins I guess. X E.
I don't need this any more, thanks for trying all. X E.
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final saddleBOT
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dawn zealot
So I'm a calc BC student and I have an epsilon-delta proofs test for continuity and limits. I am 67% sure there will be 3 questions (which are proofs) and 40 minutes at most to finish them. I'm okay with solving them but the teacher told us that at least one of the problems will require proving either the squeeze theorem, uniqueness of limits, or the limit law involving the composition of limits. I know how to prove the squeeze theorem but I hae no idea how to do the other 2 (although I guess that the uniqueness one might involve contradiction). Can someone please help me? My test is tomorrow.
final saddleBOT
So I'm a calc BC student and I have an epsilon-delta proofs test for continuity ...
opal pelican
So I'm a calc BC student and I have an epsilon-delta proofs test for continuity ...
Don't know what you mean by "limit law involving composition of limits" but I should be able to help you out with the uniqueness of the limit
dawn zealot
Thank you!
opal pelican
Alright, so, as you might've guessed, we'll prove this by contradiction
dawn zealot
Okay
So is our objective here to prove that if lim x->a f(x)=L and lim x->a f(x)=M then L=M?
opal pelican
Essentially assuming there exist two distinct limits and showing they'll be the same, the idea is that if the function converges to both limits, then it is within distance epsilon of each limit for distances lesser than certain δs, we'll choose a δ small enough to be within both of these ranges, and since this can be done for any ε>0, both limits will be arbitrarily close to each other
opal pelican
So is our objective here to prove that if lim x->a f(x)=L and lim x->a f(x)=M th...
Yup, exactly
dawn zealot
Okay. How would I proceed beyond this?
Would I suppose instead that L does not equal M
opal pelican
Would I suppose instead that L does not equal M
That's the start, yeah
dawn zealot
Then would I use the epsilon-delta definition of limits on each of the functions separately and assign a delta_1 and delta_2?
opal pelican
Then would I use the epsilon-delta definition of limits on each of the functions...
That'll be needed, thing is, as always happens in analysis, you need to choose some appropriate epsilon
Which is easy knowing the solution, not so much if you don't
dawn zealot
So would we work backwords to find epsilon?
opal pelican
So would we work backwords to find epsilon?
I don't think that'll work
Mainly because without some chosen epsilon "you have nothing"
dawn zealot
How would I proceed beyond this then?
opal pelican
You can't choose deltas and as such cannot exploit the limit condition
opal pelican
How would I proceed beyond this then?
Choosing the epsilon
The idea for this particular choice is that we want something smaller than the distance between the limits
That is, something positive but smaller than |L2-L1|
For convenience, and without loss of generality, assume L2>L1
opal pelican
So would epsilon be less than |L2-L1|
Yeah, but you need to set a certain one
That is, say ε=something
dawn zealot
Epsilon = L2-L1?
opal pelican
Epsilon = L2-L1?
That is exactly equal to your distance
What ways do you know to reduce the value of something while keeping it positive?
dawn zealot
Triangle inequality?
opal pelican
Triangle inequality?
That's mainly used to set bounds on absolute values, also, an inequality
So won't give you a valid choice of epsilon
opal pelican
L2-L1 is not a fraction though?
opal pelican
Dividing by integers
(L2-L1)/n ≤ (L2-L1), strict for n>1
And that doesn't change the sign
dawn zealot
Oh okay. Would I choose some random interger?
opal pelican
In general, when you want to find a positive value lower than another one you're given, dividing by some positive integer should be your go to choice
opal pelican
Oh okay. Would I choose some random interger?
Yeah, in this case it works out better if you choose it greater than 2
dawn zealot
okay
opal pelican
As if you choose (L1-L2)/2, L1+ε and L2-ε will "almost overlap"
dawn zealot
Would 3 work?
opal pelican
Picture it as a line segment separating them both, of length |L1-L2|
dawn zealot
Thank you
opal pelican
So well, now you have your ε, (L2-L1)/3
dawn zealot
Would this epsilon be for both |f(x)-L1| and |f(x)-M|
opal pelican
Would this epsilon be for both |f(x)-L1| and |f(x)-M|
Yeah, because you're interested on "getting L1 and L2 close", which is why we chose our epsilon based on the distance between them
And you'll need your deltas for this chosen epsilon
dawn zealot
Am I allowed to use the epsilon for both of them or do I have to explain/justify this
opal pelican
Am I allowed to use the epsilon for both of them or do I have to explain/justify...
Nothing wrong with that
The definition states something happens for any choice of epsilon
dawn zealot
Okay
opal pelican
So it in particular will work for that choice of epsilon, in both cases
dawn zealot
So with that epsilon, what should I do?
opal pelican
So with that epsilon, what should I do?
Write out the definition for that chosen epsilon, what would we want to do?
dawn zealot
|f(x)-L1|<(L2-L1)/3 and |f(x)-M|<(L2-L1)/3
opal pelican
|f(x)-L1|<(L2-L1)/3 and |f(x)-M|<(L2-L1)/3
That is part of the definition
dawn zealot
|x-a|<delta_1, |x-a|<delta_2
opal pelican
The whole thing is:
∃ (δi) such that |x-a|<δi implies |f(x)-Li|<(L2-L1)/3
dawn zealot
How would I relate the epsilon to the delta?
opal pelican
In other words: there exists some interval radius δi, such that if x is in (a-δi,a+δi), then f(x) is in (Li-ε,Li+ε), where ε is (L2-L1) in our particular case
That is, your deltas are the radius of intervals that "force your function inside other intervals", for intervals as small as you'd like
opal pelican
I still don't really know how to make the delta equal some epsilon
Because that's not what you need to do in this case, that won't help you
opal pelican
We know that within some interval (a-δ1,a+δ1) something happens, and that within some other interval (a-δ2,a+δ2) something happens as well, they're both intervals centered on the same point, so you can verify that one will contain the other
Now, we want what happens in those intervals to happen at the same time, that is, the function to be within ε of both L1 and L2
dawn zealot
Okay. Thank you for your help! I have to go eat dinner now so sorry to not be able to finish this. Thank you for helping me start and understand this more
Have a nice day
.close
burnt timber
on a function f(x) it's concave up if f''(x) is larger than 0 and cc down if f''(x) is smaller than 0 right? or is there more to this
final saddleBOT
on a function f(x) it's concave up if f''(x) is larger than 0 and cc down if f''...
burnt timber
and the inflection points are when f''(x) changes concavity?
junior token
and the inflection points are when f''(x) changes concavity?
Yes
junior token
on a function f(x) it's concave up if f''(x) is larger than 0 and cc down if f''...
You can consider x^2 to check the signs
if you ever forget which is which
junior token
on a function f(x) it's concave up if f''(x) is larger than 0 and cc down if f''...
and this is it
next thorn
d/dx (sec(2x^2 + 2x))
sec(2x^2 + 2x)*tan(2x^2+2x)
and then, times the derivative of the angle itself... my question is, how many times?
obtuse grail
what
glossy zephyr
If your question comes from the fact of using the derivative of 2x^2 + 2x twice, no
glossy zephyr
chain rule states: $\dv x f(g(x)) = f'(g(x)) \cdot g'(x)$
soft zealotBOT
next thorn
If your question comes from the fact of using the derivative of 2x^2 + 2x twice,...
ig that makes sense bc the original angle is 1.
glossy zephyr
Here g(x) is the whole 2x^2 + 2x
final saddleBOT
final saddleBOT
Available help channel!
sacred night
Factorise $x^{12}-y^{12}$
final saddleBOT
Factorise $x^{12}-y^{12}$
sacred night
!status
final saddleBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
soft zealotBOT
calvin
sacred night
is it $(x^6 + y^6)(x^6-y^6)$
soft zealotBOT
calvin
tribal crescent
is it $(x^6 + y^6)(x^6-y^6)$
u can still go on
sacred night
im confused tho
why cant we do x^6 + y^6
WIt but
Wait
X^6 wouldnt work
Would it?
Wair nvm
Im stupid
ty
.close
final saddleBOT
tribal crescent
what are you asking bro
sacred night
I forgot my exponent laws for a second
candid pulsar
why cant we do x^6 + y^6
yeah you need to go even further
few more steps
use $x^3+y^3 = (x+y)(x^2-xy+y^2)$ and $x^3-y^3 = (x-y)(x^2+xy+y^2)$ to further simplify $x^6-y^6$ and $x^6+y^6$
sacred night
How
soft zealotBOT
candid pulsar
$x^6+y^6 = (x^2)^3 + (y^2)^3$
soft zealotBOT
candid pulsar
and $(x^2)^3 + (y^2)^3 = (x^2+y^2)((x^2)^2 - (x^2)(y^2) + (y^2)^2)$ according to that formula
soft zealotBOT
candid pulsar
im guessing you are asked to factorize to the maximum possible extent and not just once
final saddleBOT
Available help channel!
mint mural
How to know if you can factor something or not?
final saddleBOT
How to know if you can factor something or not?
craggy plank
How to know if you can factor something or not?
Any example?
craggy plank
what Ann said 
pliant shore
what Ann said <:this:489575168314966026>
they were just doing a problem on quadratics, btw
craggy plank
okay, I just checked your other thread
Looks like you’re trying to factorize polynomials
craggy plank
they were just doing a problem on quadratics, btw
ye, just notice that
mint mural
For example can you factor
x^2 + x + 4
cosmic perch
$ax^2+bx+c = a(x-x_1)(x-x_2)$
soft zealotBOT
Labyrinth
cosmic perch
where x1 and x2 are the roots of x
This always works
However, I'm guessing you're restricted to real numbers, will have to check
pliant shore
I mean if you allow irrational and imaginary roots, any quadratic is factorisable
if you want only real roots, the condition is discriminant >= 0
if you want only rational roots, the condition is discriminant = perfect square
mint mural
I mean like this
pliant shore
oh, those
basically that's about if you can match it to the given form
say if you have 8z^3 - 125
that's (2z)^3 - 5^3, so you can use the form for a^3 - b^3
craggy plank
perfect squares and cubes
mint mural
Do the first one please
final tangle
wdym by do the first one
leaden moon
For example can you factor
x^2 + x + 4
for any polynomial $ax^2 + bx + c$, consider the discriminant $\Delta = b^2 - 4ac$
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
if $\Delta \geq 0$, it's factorizable
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
else, you can't factor it
specifically, if $\Delta = 0$, you can turn your quadratic into a perfect square
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
@mint mural
lunar jewel
mint mural
I got 3 + sqrt(11) and 3 - sqrt(11)
mint mural
those are fucking zero points
final tangle
get their decimal approximation
put a scale on your graph
and put a point around that approximation
mint mural
Ok @final tangle
final saddleBOT
@mint mural Has your question been resolved?
final saddleBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
leaden moon
because it seems like im confused of what you are trying to ask
firstly, you asked when is a polynomial factorizable, secondly, you go on with your zero points
what question are you trying to encounter?
mint mural
.close
versed dawn
Hi, is my solution right? Probability item 🥲
Mia draws two pens one after another from a box that contains 5 red pens and 6 blue pens, without putting any pen back. Let X be the random variable that indicates how many blue pens she draws. Determine the possible values of X. Compute the mean of the probability distribution.
pliant shore
Hi, is my solution right? Probability item 🥲
Mia draws two pens one after anot...
your probabilities are wrong
given there are 5 red pens and 6 blue pens, what's the probability Mia draws a red pen?
versed dawn
is it 5/11
pliant shore
is it 5/11
yep!
alright, and we take the red pen out, so now there are 4 red pens and 6 blue pens
versed dawn
wait is the first or second table wrong
pliant shore
wait is the first or second table wrong
the 2nd table
versed dawn
alright, and we take the red pen out, so now there are 4 red pens and 6 blue pen...
so how do i put this down in the table tho..
cuz what i did, i based it off the first table
pliant shore
alright, and we take the red pen out, so now there are 4 red pens and 6 blue pen...
can you answer this for the 2nd red pen first?
once you get the logic for RR (X = 0) the rest will follow
versed dawn
can you answer this for the 2nd red pen first?
4/11
but arent we looking for the blue pen tho
versed dawn
then simplify
versed dawn
but arent we looking for the blue pen tho
.
pliant shore
but arent we looking for the blue pen tho
I don't know what you mean
versed dawn
Hi, is my solution right? Probability item 🥲
Mia draws two pens one after anot...
Let x be the random variable that indicates how many blue pens she draws.
Determine the possible values of X. Compute the mean of the probability distribution.
pliant shore
Let x be the random variable that indicates how many blue pens she draws.
yes, so I'm correcting you on 'compute the mean of the probability distribution'
you've done 'determine the possible values of X' correctly
well you just needed to write X = {0, 1, 2} but your table is fine
versed dawn
how do i do the 2nd table tho, sorry i just based it off my teachers lesson
pliant shore
how do i do the 2nd table tho, sorry i just based it off my teachers lesson
do you agree that P(RR) = 5/11 * 4/10 then?
you need to find P(RR), P(RB), P(BR), P(BB) first
versed dawn
do you agree that P(RR) = 5/11 * 4/10 then?
ye
versed dawn
you need to find P(RR), P(RB), P(BR), P(BB) first
ohh
then separate tables for that in the format of 2nd table?
pliant shore
then separate tables for that in the format of 2nd table?
then yes, P(X = 1) = P(RB) + P(BR)
once you find the correct probabilities, then you can do 0 * ... + 1 * ... + 2 * ...
you can't just assume the probabilities are 1/4, 2/4, 1/4
versed dawn
okii, ill write them down first
versed dawn
you can't just assume the probabilities are 1/4, 2/4, 1/4
Like this?
pliant shore
Like this?
you don't need to multiply by 2/11 or 3/11 at the end of each one
pliant shore
otherwise, that's correct
versed dawn
once you find the correct probabilities, then you can do 0 * ... + 1 * ... + 2 *...
What do i place in the blanks if theres 4 outcomes but only 3 values
Do i repeat the 1
pliant shore
What do i place in the blanks if theres 4 outcomes but only 3 values
as I said before, P(X = 1) = P(RB) + P(BR)
versed dawn
Ohhh
pliant shore
,calc (5/11 * 6/10 + 6/11 * 5/10) + 2 * 6/11 * 5/10
soft zealotBOT
Result:
1.0909090909091
Oh yeah
pliant shore
we're just looking for our expected number of blue pens to be between 0 and 2
viral jungle
minor nitpick: is P(X) supposed to be a probability or an expected value?
pliant shore
it actually makes sense that our answer is near 1, actually
if we make it 5 red pens and 5 blue pens
and we do it with replacement
then that's just the setup of a fair coin
pliant shore
minor nitpick: is P(X) supposed to be a probability or an expected value?
indeed, they should write E(X)
versed dawn
So 1.09 is the mean?
viral jungle
right, sorry to interrupt
pliant shore
So 1.09 is the mean?
yes
versed dawn
No prob
pliant shore
right, sorry to interrupt
not at all, I didn't catch that
wasn't looking carefully enough haha
versed dawn
I get it noww, tyzm for teaching my slow brainer 🥹
pliant shore
no worries!
versed dawn
no worries!
the 2nd table isnt needed anymore right?
pliant shore
the 2nd table isnt needed anymore right?
yeah, we fixed everything in the 2nd table
it's not needed
versed dawn
okaayy
pure cliff
final saddleBOT
viral jungle
have you tried sketching the problem?
viral jungle
well, C has to lie on the line in order to divide AB, yes?
pure cliff
so ill just plot it at a random place
viral jungle
and given AB as a line segment and C dividing it in the ratio 1:3, which side do you think would be longer: AC, or BC?
pure cliff
Bc
viral jungle
where's the diagram?
viral jungle
right.
so, you are given that BC:AC = 3:1.
have you learnt about the distance formula?
pure cliff
ye
viral jungle
if you have, you can apply it here.
use the fact that the distance from C to B is 3 times that of A to C
let the coordinates of B be (x, y), then solve for that
or, to simplify, you can solve one component (x or y) at a time
which is easier and less headache to track
viral jungle
ack, might be overcomplicated here
rocky girder
you can use this formula where
B(x2,y2) A(x1,y1)
viral jungle
actually, just do it component-wise. I think it's easier that way
rocky girder
it gives you a point in a ratio
viral jungle
yeah this works
rocky girder
try and use that one and see if that works
rocky girder
nice
pure cliff
thanks im prob gonna need more help in a few mins
sharp wraith
find all functions that satisfy f(2x) = f(x) for all positive integers, how are you meant to approach this, i found that f(1) = f(2) = ... = f(2^k), and that f(3) = f(2^k 3), therefore f(2k+1) = f(2^k(2k+1))
final saddleBOT
find all functions that satisfy f(2x) = f(x) for all positive integers, how are ...
tired walrus
that feels like either underspecified or a really wide class of functions 
lucid nymph
It would probably have to be either a constant or a sinusoidal function since it only means for all positive integers
final saddleBOT
final saddleBOT
Available help channel!
supple nest
2
supple nest
Making sure my answer suffices
For how nasty a lot of proofs are this one seems pretty simple
shadow marlin
What is "a" for ?
final saddleBOT
@supple nest Has your question been resolved?
boreal lichen
dyadic folding identity
distant scarab
,, \sum_{k=1}^\infty \frac{n^2}{n+k}
final saddleBOT
,, \sum_{k=1}^\infty \frac{n^2}{n+k}
distant scarab
how to calculate this limit
soft zealotBOT
vital crag
looks like it diverges for most n
distant scarab
wait idk if what i wrote is correct
its $\lim_{n \to \infty} \sum_{k=1}^n \frac{n}{n^2+k}$
no way i wrote it inverted
soft zealotBOT
distant scarab
@rustic sequoia
atomic moon
Try to write it as a riemann sum
distant scarab
Try to write it as a riemann sum
which one
atomic moon
Never mind you can brutally make an asymptotical equivalent of the term inside
cancel n on top and bot
distant scarab
cancel n on top and bot
i dont get it
atomic moon
You agree this is the same as : n/n²(1+k/n²)
So this is 1/n(1+k/n²)
Look at k/n² for k in {0,...,n} and n tends to +inf
Think and tell me
distant scarab
Look at k/n² for k in {0,...,n} and n tends to +inf
its 0
floral nova
If you know enough complex analysis, the way I would approach it is to use the difference equation $$\psi(x+N+1)-\psi(x) = \sum_{k=0}^{N} \frac{1}{x+k}$$. You can then use the $\psi$ to telescope it down.
soft zealotBOT
JessicaK
distant scarab
i see it now , the limit is 1
distant scarab
So what can you say about each term
huh?
it would just cancel out as n -> infinty
atomic moon
I was thinking of saying that 1/n(1+k/n²) ~ 1/n and so you sum on it and its just (n+1)/n which tends to 1 indeed
distant scarab
.close
final saddleBOT
final saddleBOT
Available help channel!
whole halo
Let G be a finite simple group. Let p be a prime dividing |G|. Let H ≤ G be the group generated by all elements of order p. Prove that G = H.
final saddleBOT
Let G be a finite simple group. Let p be a prime dividing |G|. Let H ≤ G be the ...
hasty mist
What have you tried?
whole halo
the only thing I can think of is that I need to prove that H is a normal subgroup of G, other than that Ive got nothing
scarlet sequoia
the only thing I can think of is that I need to prove that H is a normal subgrou...
that's exactly it
whole halo
oh goddammit
I crossed that out because I didnt do it right
.close nevermind on this question too then
final saddleBOT
whole halo
absolutely bombed the final exam because nothing I could think of seemed to work
I didnt even try and work with the ideas I had
thats 2 out of 3 questions I couldve finished had I stuck with them for longer
spent an entire half of the exam time staring at those 3 questions
final saddleBOT
Available help channel!
digital wave
If A is a group, is (A)^n also a group? What about (Z2)^n?
final saddleBOT
If A is a group, is (A)^n also a group? What about (Z2)^n?
digital wave
Group with the group set being Cartesian producted with itself n times
trail mango
okay
digital wave
The group operation becomes the original group operation of all individual elements from the sequence
trail mango
do you think it's a group?
digital wave
(a1, a2,...an)#(b1,...bn) = (a1b1, ..., anbn)
trail mango
do you need to prove it?
digital wave
No, I just want to know if it's true in general or not
digital wave
Oh
trail mango
that operation is called direct product of groups
trail mango
usually you would write it like $$\prod_{i=1}^n A$$
soft zealotBOT
slayla
final saddleBOT
trail mango
no problemo
final saddleBOT
✅ Original question: #help-36 message
digital wave
So like
Does it fail for other structures
What about rings ornfields
It feels like I remember it failing somewhere
opal pelican
Does it fail for other structures
Can't think of any structure the direct product fails on
digital wave
opal pelican
What about rings ornfields
Pretty sure it holds for rings as well, and, by extension, fields
One thing, though:
opal pelican
In the case of a field, it not necessarily preserves field structure
But it will preserve the ring one
trail mango
RxR is not a field
opal pelican
RxR is not a field
Precisely what I said above
digital wave
Can I keep the channel open for further questions or make a new channel everytime
Bcz I will have more questions
trail mango
whatever you want. but you should probably close if you won't have any new questions soon
final saddleBOT
trail mango
why do you cry all the time
trail mango
it's ok
scarlet sequoia
I think it fails for integral domains too
yes you don't keep integral property with cartesian product of rings
(a,0) * (0,b) = 0
digital wave
Yeah
final saddleBOT
Available help channel!
halcyon fog
Suppose tan theta=x and theta is in quadrant 2, such that x<0. Determine the exact values of sin theta, cos theta, sec theta, csc theta, and cot theta in terms of x. Review before my final and i got i got sin =-x/root x^2+-1^2 (abbreivated from now on as z). sec= z/-1. cos=-1/z. csc=z/x. cot= -1/x. I used an A.I to fact check and its just confusing me. Since its quadrant 2 x is negative so x/-1 as sin is positive in quadrant 2 and cosine is negative, the A.I is saying cot theta= 1/x but isnt it -1/x?
final saddleBOT
Suppose tan theta=x and theta is in quadrant 2, such that x<0. Determine the exa...
final saddleBOT
final saddleBOT
Available help channel!
heavy sparrow
final saddleBOT
craggy plank
I don’t know the complete solution to the entire question, but can provide you some information regarding certain options
heavy sparrow
I tried jensen
But,I am confused when the double derivative sign changes in second quadrant
shy mantle
I tried jensen
wus tha
heavy sparrow
Jensen inequality
final saddleBOT
@heavy sparrow Has your question been resolved?
silver dew
You are on the right track
cos,sin,tan, one of them is concave finction
And (A+B+C)/3 is constantly π/3
One of them is concave on [0, π] I meant
silver dew
Do you see which one?
Interesting, Jensen leads you to one correct option, but actually two options are correct
silver dew
Sum of cos = 1+(radius of inner circle/radius of outer circle) and the last term has a range (0,1/2].
you need to use this to deduce the another correct option. If this requires you to select only one, select the one Jensen leads you. The person gave you this exercise probably didn’t expect you to come up with two.
proven radish
I did this in my jee time
same
final saddleBOT
final saddleBOT
Available help channel!
spark swan
do we really need to use MVT?
can't we make |f_n(x)-f_n(x_0)| and |f_m(x)-f_m(x_0)| arbitrary small by uniform continuity, since each f_n is differentiable which implies continuity and [a,b] is compact so it's uniformly continuous
and |f_n(x_0)-f_m(x_0)| arbitrarily small because f_n converges at x_0
f'=g because of another already proved theorem (basically this theorem but f is assumed to converge pointwise)
final saddleBOT
do we really need to use MVT?
can't we make |f_n(x)-f_n(x_0)| and |f_m(x)-f_m(x_...
@spark swan Has your question been resolved?
gloomy river
you can only make |f_n(x) - f_n(x_0)| small if x is close to x_0
silver dew
Then you shouldn’t consider it alone, but two , m and n together
$(f_{n}(x)-f_{m}(x))-(f_{n}(x_{0})-f_{m}(x_{0}))=\int_{x_{0}}^{x}(f_{n}’(t)-f_{m}’(t))dt$
Just as the hint said
soft zealotBOT
Cogwheels of the mind
silver dew
|x-x_0|<=b-a and the thing in the integral can be really small
gloomy river
is this about what I said?
spark swan
you can only make |f_n(x) - f_n(x_0)| small if x is close to x_0
right i realised my goof
spark swan
|x-x_0|<=b-a and the thing in the integral can be really small
i don't get how to how the integral can be made small 😭
silver dew
f_n’ converge uniformly, so f_n’ is Cauchy uniformly
spark swan
f_n’ converge uniformly, so f_n’ is Cauchy uniformly
oh yeah
so i just set the integrand to be less than epsilon/(b-a)
thank you @silver dew and @gloomy river
.close
distant scarab
,, \lim_{x \to 0} \left( \frac{2}{\sin(x)^2} - \frac{1}{1-\cos(x)} \right)
soft zealotBOT
frosty sigil
Simplify it into 1 eq first, it might help
Sin²x can be 1-cos²x
1-cos²x => (1+cosx)(1-cosx)
Then you should be able to make it into 1 eq then see how to go bout it starting from there
distant scarab
Then you should be able to make it into 1 eq then see how to go bout it starting...
i got $\frac{1-\cos(x)}{\sin(x)}$
soft zealotBOT
distant scarab
,, \left( \frac{2}{\sin(x)^2} - \frac{1}{1-\cos(x)} \right) = \frac{1-2\cos(x) + \cos(x)^2}{(1-\cos(x)) \cdot \sin(x)^2} = \frac{((1-\cos(x))^2}{(1-\cos(x)) \cdot \sin(x)^2}
soft zealotBOT
frosty sigil
still get 0/0
Great, use lhopital
wraith tide
i am think it must be this
distant scarab
Great, use lhopital
not allowed
distant scarab
i am think it must be this
so the limit is infinty ?
frosty sigil
Hmm the way you simplified it is off
wraith tide
so the limit is infinty ?
Yes, it was my mistake. It should have been there 1/(1 + cox(x))
distant scarab
Hmm the way you simplified it is off
how
soft zealotBOT
coarse coral
how to evaluate this limit
wraith tide
distant scarab
i mean this isnt the same as the limit that im asking for
its 1-cos(x) in the denominator
frosty sigil
sin²x can be 1-cos²x which can be (1+cosx)(1-cosx) then the (1-cosx)s can be cancelled out
frosty sigil
Hmm I'm pretty much sure this is under !noans
coarse coral
i mean this isnt the same as the limit that im asking for
oh, I've forgot to write "1-", but when you will solve this you will get the same answer because I solved this for 1-cosx in denominator
frosty sigil
!noans
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distant scarab
sin²x can be 1-cos²x which can be (1+cosx)(1-cosx) then the (1-cosx)s can be can...
thx
.close
final saddleBOT
oops sorry i don't knew it ty
wraith tide
.close
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distant scarab
$\lim_{x \to 0} \frac{e^{2x} +5e^x -6}{e^x-1}$
soft zealotBOT
final saddleBOT
$\lim_{x \to 0} \frac{e^{2x} +5e^x -6}{e^x-1}$
loud sundial
factor
tired walrus
first and foremost
why not substitute y := e^x
it's practically begging to be made
$\lim_{y \to 1} \frac{y^2+5y-6}{y-1}$
soft zealotBOT
Ann
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tranquil pine
hiiii could i get some help on this pls
final saddleBOT
hiiii could i get some help on this pls
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tbh idk where to begin
shoudl i attempt proof by induction
pls help me 😭
knief cmon 😭
rocky tusk
do you think it’s true?
rocky tusk
ok so what’s the issue
tranquil pine
how do i prove it
rocky tusk
did you use the definitions they gave above?
tranquil pine
ngl i didnt
tranquil pine
shoudl i attempt proof by induction
is this the right idea