How
#help-0
flint zealot
pine cave
japan
can you send a picture of your kb
pine cave
alr sure
pine cave
wdym its japan layout
its hard to find guys quit
i also need help with
<@&286206848099549185> ?
half patrol
brochacho wheres the japan layout 🥀
pine cave
that one?
half patrol
is this not just a keyboard
pine cave
it is sorry
vale wigeon
ok so do you know how to solve quadratic equations
pine cave
nop im new to that
vale wigeon
that would be best, yes
vale wigeon
mate you definitely have a minus sign on your keyboard
pine cave
i got I
vale wigeon
mmm anyway can you show your work for how you got these
pine cave
oh i got P
vale wigeon
zero, not O
pine cave
i have it my handwriting
vale wigeon
right now all i can say is that it could be that the equation really does have 4 solutions, but we would need to check that your work is correct.
vale wigeon
,rcw
ocean sealBOT
pine cave
btw my handwriting good?
vale wigeon
your handwriting is readable
vale wigeon
no, i am not a kid either.
vale wigeon
indeed, i don't think that 26 years old counts as a "kid".
except maybe if you're in your seventies
pine cave
sorry miss
How to deal with minus signs while simplifying expressions?
it's always confusing
pine cave
nice hand writing
solve plz
half patrol
was there a specific question that prompted this or is this a general doubt
vale wigeon
How to deal with minus signs while simplifying expressions?
thats like saying "how to deal with the number 4 while simplifying expressions" 💔
half patrol
wait before i say anything is it the subtraction and minus part that gets to u
like the wording of them?
half patrol
its a really general question sorry g
pine cave
some tricks?
half patrol
im assuming u meant something along the lines of (-x) * (-2) for example
pine cave
yea THS KEYBAORD ANNOYING AS HELL
half patrol
well as long as you know
negative * negative = positive
positive * negative = negative
you should be fine
pine cave
yeah thats it
half patrol
ohhhh yea got u bro
pine cave
close this ig im done
half patrol
😭😭
half patrol
.close
half patrol
id love to but i cant
vale wigeon
let me put y'all out of your misery
pine cave
then what do i do?
vale wigeon
.close
lone heartBOT
pine cave
thanks
vale wigeon
it's a dot not a comma
pine cave
.close
vale wigeon
already closed
pine cave
oh
vale wigeon
it'll release in like 10 min
pine cave
k
pine cave
it's a _dot_ not a comma
oh lol
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pallid crystal
Can someone explain what No Elementary Anti-derivative means?
I was trying to integrate e to the power of x squared
pallid crystal
Is it similar to saying that it has no solution in algebra
late sage
.
regal totem
Can someone explain what No Elementary Anti-derivative means?
I was trying to in...
That means, no combinations of the functions we commonly use (polynomials, exponentials, logarithms, trig functions) upon differentiation willl yield e power x squared
hushed locust
it means that an antiderivative exists, but there is no formula for it in terms of exponentials, trig, log, polynomials, roots, etc (plus addition,multiplication, subtraction, division and composition of the above)
regal totem
Is it similar to saying that it has no solution in algebra
no, "no solution" means there is no such x in some set such that x satisfies the property
pallid crystal
I think what I'm confused on is how the antiderivative exists if the function doesn't give e power x square when differentiating
I'm not sure I'm expressing it correctly, sorry
Shouldn't a function be differentiable in order to have an anti derivative?
hushed locust
Shouldn't a function be differentiable in order to have an anti derivative?
no, not necessarily
limpid turret
<@&268886789983436800> self endorsement
clever folio
No. Don't use this server to try and game the arxiv endorsement system.
clever kite
sly cobalt
Can someone help me with this problem?
lone heartBOT
Can someone help me with this problem?
limpid turret
Can someone help me with this problem?
Pythag
sly cobalt
bc pythag is a^2+b^2=c^2
sly cobalt
bc thats what one of the lengths are
limpid turret
It is?
sly cobalt
idk
limpid turret
The radius of the circle is 13
ivory igloo
now use pythagoras
you know lengths of the arms of the right triangle to be 26 and 9 now
sly cobalt
what would the equation set up be
tulip flame
what would the equation set up be
what you think?
tulip flame
am i missing something?
go on
tulip flame
now do we square tha?
take square root $\sqrt{757}$
sly cobalt
.
ocean sealBOT
heisenberg
sly cobalt
how did i forget that
ocean sealBOT
The following error occured while calculating:
Error: Unexpected operator { (char 5)
tulip flame
,w sqrt757
ocean sealBOT
tulip flame
: calculatorman
lone heartBOT
If you are done with this channel, please mark your problem as solved by typing .close
sly cobalt
.close
lone heartBOT
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tame nebula
lone heartBOT
tame nebula
am i doing this right so far
this for finding radius of convergence
also why wouldnt it be
0
instead of the |x|
on this solution here the R is 1
why is it 1? and what would the interval be
lone heartBOT
@tame nebula Has your question been resolved?
upbeat gorge
The way you cancel stuff is…a bit weird
tame nebula
The way you cancel stuff is…a bit weird
is it wrong
upbeat gorge
So first line, ratio test, no problem
Next line though, your solution seems to magically put new numbers for the sake of cancelling them
Rather than working with the originals
For example, $\frac{x^{n+1}}{x^n}\neq \frac{x^{n+1}x^n}{x^n}$
ocean sealBOT
CST (reply ping for help)
upbeat gorge
is it wrong
So yeah, please recheck
tame nebula
tame nebula
So yeah, please recheck
think i go it
upbeat gorge
Have to put the limit thing since you still haven’t taken the limit for |(n+1)/(n+2)|
tame nebula
ik
tame nebula
Have to put the limit thing since you still haven’t taken the limit for |(n+1)/(...
but is the cancelling right now
upbeat gorge
Yep
upbeat gorge
What are the conditions for the ratio test to say that a series converges?
tame nebula
What are the conditions for the ratio test to say that a series converges?
must be less than 1
tame nebula
|x|
upbeat gorge
And sooo
tame nebula
if its 1 it cant be because it diverges?
upbeat gorge
If it’s equal to 1, the ratio test is inconclusive
You’ll have to use a different test
tame nebula
oh yea right forgot about that
tame nebula
if its 1 it cant be because it diverges?
so is this right except its just inconclusive instead of diverges
upbeat gorge
Try to see if you can conclude something
What other convergence tests do you know of
Or I guess an alternate question is, can you recognize the series if x=1 or x=-1?
alpine sable
how do i know if its B or C? helppp
lone heartBOT
how do i know if its B or C? helppp
nova glen
the diagram shows that the angle is in the second quadrant
cosine is always negative in the second quadrant
lone heartBOT
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subtle cedar
For the LHS, can I cancel out the (2n-2)! in the denominator and just leave 2n and 2n-1 in the top?
subtle cedar
ok sweet lets see how far i get now
Oh W
I think I did this one all on my own except for that double check
buoyant saddle
this is incorrect
buoyant saddle
n(2n - 1) = 2n^2 - n likewise on the right you have n(n - 1) = n^2 - n not n^2 - 1
change the 1s to n's and you'll be good
subtle cedar
Wait what
subtle cedar
Oh I see
buoyant saddle
this is incorrect
buoyant saddle
yea
cinder sundial
Jesse!
mossy field
what? can you give the whole thing lol
cinder sundial
It is the whole thing
zinc haven
Is it a line or a plane in a space
intuitively, with a plane you have two "freely moving variables", whereas for a line you have one
cinder sundial
Ppl doubt me all the time. Modern ppl are so rude
mossy field
eb [explain better]
i dont even know what space you're talking about lol, 2D or 3D or (for god's sake) 4D?
zinc haven
looks like a line to me
lone heartBOT
@cinder sundial Has your question been resolved?
lone heartBOT
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queen ginkgo
lone heartBOT
queen ginkgo
Can anyone help me figoure out this..?
I have been stuck on this for a while
its a basic question,
its understanable, but i am having trouble with the proof
lone heartBOT
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.
tawny crown
Its not incomplete imo
For polynomials, You can split any polymomial into terms of two types
x^even and x^odd
Regardlessn , part b is the general even od decomposition
tawny crown
Now x^even is an even function
queen ginkgo
ok
tawny crown
You do know what "even function" means right?
queen ginkgo
You do know what "even function" means right?
i know what they mean
when f(x) = f(-x)
tawny crown
Yes, a polynomial is nothing but an expression of the form a0x⁰ + a1x + a2x²+...
Youre splitting this sum
Into two sums
queen ginkgo
i know that
thats understandable
but what i am looking for is
how can i algebrically prove that
not through substiution
tawny crown
I'm not telling you to do any substitution
Just rearrangement
And regrouping of terms
queen ginkgo
u mean
like take a even function and odd function and just add them?
and conclude that it is a polynomial?
I am looking for a proof
Is there anyone else?
pearl lake
Which one
tawny crown
and conclude that it is a polynomial?
This is a general polynomial, that IS how they always look. There are no assumptions/random conclusions
queen ginkgo
queen ginkgo
This is a general polynomial, that IS how they always look. There are no assumpt...
Hmm
pearl lake
Ok
queen ginkgo
This is a general polynomial, that IS how they always look. There are no assumpt...
Did you meant like adding them together as in regrouping?
Can you be more precise?
tawny crown
I am not adding anything new mate
The thing is already there
Im just using commutativity
Of addition
And associativity
Take for example 3x³+5x²+1
queen ginkgo
but how would that exactly prove the question tho?
Oh i take it
U have taken two polynomails
one is a even and other is odd
tawny crown
but how would that exactly prove the question tho?
The left bracket is an even function
The right is an odd function
queen ginkgo
that i understood
tawny crown
U have taken two polynomails
Quite the opposite
I started with ONE polynomial
And I showed I can write that as a sum of 2 polynomials
pearl lake
Do you know hot to write general way of writing even and odd functions
Literally just add those two up
Done in a single step
queen ginkgo
And I showed I can write that as a sum of 2 polynomials
This helps thanks
Was much simpler than expected
Should have paid more attention.
Again, thanks a lot.
.close
lone heartBOT
bronze salmon
This is a general polynomial, that IS how they always look. There are no assumpt...
Use this
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Available help channel!
low dust
lone heartBOT
low dust
no idea where to begin
a
intermediate value theorem, extreme value theorem, preservation of compact and connected sets, sequential criterion for continuity
intermediate value theorem?
another exercise was showing that if the values at the endpoints are equal, and the function is continuous in the interval, the set of discontinuities is uncountable which i was able to do
1/2+x
I'm not sure
it should be continuous
it has to be negative somewhere and positive too?
cinder compass
also.. what's the domain of g?
low dust
0 and 1/2?
low dust
also.. what's the domain of g?
has to be continuous over [0,1/2]
g(0)=f(0)-f(1/2)=-g(1/2)
crimson trout
Scroll dowb then
pearl lake
This?
low dust
g(1/2)=f(1/2)+f(1)
f(1)=f(0)
so it becomes
f(0)-f(1/2)-f(0)
oh i think i messed up
oh right
thank you!
.close
lone heartBOT
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visual eagle
can someone explain this
lone heartBOT
can someone explain this
visual eagle
and thhis
winter light
can someone explain this
Hint: similar triangles
tulip flame
i still dont get first one
you know tantheta = opp/adjacent
visual eagle
yes
pseudo ice
you know tantheta = opp/adjacent
You can do the question without reference to the angle theta, but nonetheless, you probably don't want to use sin (which also will reveal why you don't need to refer to the angle)
tulip flame
You can do the question without reference to the angle theta, but nonetheless, y...
mb
deep dune
lone heartBOT
@visual eagle Has your question been resolved?
lone heartBOT
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mortal crow
Uni Taylor Polynomials - I don't really understand what I could have wrong here - I'm going off the general formula of f(c) + f'(c)(x-c) + f''(c)(x-c)/1! ...... f(n')(c)(x-c)/n! -- am I misunderstanding what a 3rd-order taylor polynomial would be?
tacit arch
show all your derivatives
mortal crow
f(x) = (3x+46)^4/3
f'(x) = 4(3x+46)^1/3
f''(x) = 4(3x+46)^-2/3
f'''(x) = -8(3x+46)^-5/3
tacit arch
oh you're missing powers on (x-6) in your formula
you can also simplify some numbers but they're not wrong
,calc 4 * (3 * 6 + 46)^(1/3)
ocean sealBOT
Result:
16
no
tacit arch
,calc 4 * (3 * 6 + 46)^(1/3)
this is f'(6). why do you have 256^(1/3)
mortal crow
oh I think I was doing (4*(3x+46))^1/3 instead of 4(3x+46)^1/3
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Available help channel!
lone heartBOT
molten tiger
uhh...this is not how you use a help thread lmao
Sorry ?
molten tiger
Just use it to ask questions, that question was already resolved
Oh ok
lone heartBOT
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prime kindle
Im kinda clueless on how to solve this. We've been doing division of polynomials first by using long division on them and now Horner's algorithm (or synthetic division). From what i got (due to the polynomial remainder theorem where $f(p)=r$) $f(-2)=4$ and $f(5)=3$. My problem now is I don't know how to combine these two. Do i say that $f(-2\cdot5)=4\cdot3$? Or maybe something else?
ocean sealBOT
Gopher
prime kindle
I don't think we can do anything with f as it could be of degree 1, 2, or a billion
(Oh also i dont think i need to mention this but (x^2-3x-10)=(x+2)(x-5))
molten pivot
This seems like a CRT problem but with polynomials
errant zinc
the hell is horner's algorithm
prime kindle
Uhhhh
errant zinc
is that where you use bus stop method
shut kite
Im kinda clueless on how to solve this. We've been doing division of polynomials...
What's the form of the remainder when you divide f by (x²-3x-10) ?
errant zinc
is that where you use bus stop method
this thing
prime kindle
No not that
errant zinc
ok nvm lol
prime kindle
What's the form of the remainder when you divide f by (x²-3x-10) ?
I would assume it would be linear since x² is the highest degree
errant zinc
ive had a look, its very similar to bus stop
shut kite
I would assume it would be linear since x² is the highest degree
exactly
So you would have f(x) = (x²-3x-10) * Q(X) + (ax+b) then ?
with Q(X) being some polynominal
prime kindle
ive had a look, its very similar to bus stop
This kinda stuff
prime kindle
So you would have f(x) = (x²-3x-10) * Q(X) + (ax+b) then ?
Yeah
prime kindle
We have to find (ax+b)
shut kite
knowing what's f(5) and f(-2) equal to that should be pretty easy
shut kite
So you would have f(x) = (x²-3x-10) * Q(X) + (ax+b) then ?
always using that very relation ofc
bronze salmon
This kinda stuff
Is this rational root theorem
Using P/Q to determinate their factor
prime kindle
Wait wait
Does this mean that in f(-2) and f(5) the Q function is the same?
NVM DOESNT MATTER THEY CANCEL OUT
shut kite
NVM DOESNT MATTER THEY CANCEL OUT
And now you see why its the good way to do it hehe
In the end you should just get a system of two equation to solve for a and b
lone heartBOT
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coarse crag
how to solve for k?
lone heartBOT
how to solve for k?
coarse crag
you have a constraint on that
region 1 is twice as big as region 2
vapid sinew
exactly
vale wigeon
!redir
lone heartBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
vapid sinew
so u know integral from 0 to k is [p] times as big as integral from k to 5
now what is p?
vapid sinew
lets allow h to answer it himself
brave pagoda
ok
vale wigeon
!nosols
lone heartBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
small lance
👀
coarse crag
0 to k=2p
k to 5 = p
vapid sinew
0 to k=2p
k to 5 = p
okay i see where u r coming from. Just to confirm, what region does p represent for you?
is it region 1 or region 2?
brave pagoda
The literal first word is "how"
seethe qeustion gives area is twice
(region 1 area) = 2. (region 2 area)
but region 1 =
= region 2
lone heartBOT
@coarse crag Has your question been resolved?
coarse crag
okay i see where u r coming from. Just to confirm, what region does p represent ...
p is region 2
coarse crag
how to solve for k?
before I proceed to b, Is q a) correct?
gray isle
Some parts are problematic
1/2 ln(51) isn't ln 2601
And missing () next to the 1/2
1/2 ( [] - [] )
coarse crag
Some parts are problematic
so up to the the second last line of question a is problematic?
Il come back to this problem
.close
lone heartBOT
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rich river
hi
rich river
wheat isle
lmfao
wheat isle
ur cooked
rich river
ember bane
k what's the definition of a cauchy sequence?
ember bane
ill give a hint
ember bane
from this equation itself, for $p>q$ , find the difference $|x_p - x_q|$
ocean sealBOT
Annie Maqionde
ember bane
didnt mean find an equality
rich river
its just the elements between them
ember bane
its just the elements between them
as an example $|x_{n+2} - x_{n} + x_{n} - x_{n-1}| \le |x_{n+2} - x_{n}| + | x_{n} - x_{n-1}|$ so on so forth
ocean sealBOT
Annie Maqionde
rich river
xp + xp-1 + xp-2.... - xq - xq-1 - xq-2...
ember bane
?
rich river
u asked the terms between xp and xq right?
ember bane
k much simpler just consider $|x_{n+p}-x_{p}$
ocean sealBOT
Annie Maqionde
ember bane
u asked the terms between xp and xq right?
not the terms the difference between the two
ember bane
**Annie Maqionde**
from the given equation, you need to find an $N$ such that $|x_{n+p} - x_{p}| \le N$
ocean sealBOT
Annie Maqionde
rich river
not the terms the difference between the two
how can i find the difference?
u mean e?
ember bane
**Annie Maqionde**
....
ember bane
well the thing is that in the definition of a cauchy sequence $e$ is givne and we need to find an $N$ such that for every $n$, $|x_{M+n} - x_n < e$
ocean sealBOT
Annie Maqionde
ember bane
so to do that we need to first let's say investigate the sequence itself
ember bane
ik but its simpler if we think it that way
ember bane
cuz then we can apply this
rich river
ohh go on
rich river
**Annie Maqionde**
wait e is given?
ember bane
yes, for givne $e$ we need to find an $M$
ocean sealBOT
Annie Maqionde
ember bane
**Annie Maqionde**
then you can use this
$ |x_{n+2} - x_{n}| = |x_{n+2} - x_{n} + x_{n} - x_{n-1}| \le |x_{n+2} - x_{n}| + | x_{n} - x_{n-1}| \le r^{n-1} + r^n$
for example
rich river
then you can use this
i got a geometric series
ember bane
exactly
now try to find an upper bound for that
its a partial sum of an infinite geoemtry series, right?
rich river
its a partial sum of an infinite geoemtry series, right?
well its was in terms m and n but we can simplify ig
rn/1-r?
0 < r < 1
0?
bound?
ember bane
so $r^n + r^{n-1}......+ r^{m} < \sum^{\infty}r^i$ right?
ocean sealBOT
Annie Maqionde
rich river
yes
ember bane
exactly
that's an upper bound
so now you have $|x_{m+n} - x_m| \le \sum^{\infty} r^i$
ocean sealBOT
Annie Maqionde
ember bane
and since $|r|<1$ then you can calculate the sum on the right
ocean sealBOT
Annie Maqionde
rich river
rn/1-r
ocean sealBOT
Annie Maqionde
ocean sealBOT
Annie Maqionde
rich river
**Annie Maqionde**
here
rich river
now set $\epsilon = \frac{1}{1-r}$
or is this correct 😭
ember bane
but we dont have the 1 term
?
ember bane
**Annie Maqionde**
here we sum $\sum^{\infty}_{i=0} r^i$
ocean sealBOT
Annie Maqionde
ember bane
?
ember bane
what I meant was $r^m + ..... + r^n < \sum^{\infty}_{i=0} r^i = \frac{1}{1-r}$
ocean sealBOT
Annie Maqionde
ember bane
k show your method
rich river
what I meant was $r^m + ..... + r^n < \sum^{\infty}_{i=0} r^i = \frac{1}{1-r}$
$r^m + ..... + r^n = \sum^{\n}_{i=m} r^i = \frac{r^n}{1-r}$
ocean sealBOT
vie
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
ember bane
yes that is conceptually correct
but it would make the whole proof that ${x}$ is cauchy a bit more difficult
ocean sealBOT
Annie Maqionde
ember bane
**Annie Maqionde**
...
ember bane
no
rich river
or we have to like prove e > 0
ocean sealBOT
Annie Maqionde
rich river
**Annie Maqionde**
wait
ember bane
what if $e$ is not of that form?
ocean sealBOT
Annie Maqionde
rich river
-1>1/r-1>0
ember bane
so what we do now is that we take an arbitrary $e$ and first prove the existence of an $r$ such that $1/(1-r) < e$
ocean sealBOT
Annie Maqionde
rich river
we can get that from 0 < r < 1
ember bane
yes so after that the proof should be complete
lone heartBOT
rich river
.reopen
lone heartBOT
✅ Original question: #help-0 message
modern sedge
Hint: prove that its cauchy
dense portal
you can try to prove this up to a constant
like use what you just proved for this one
modern sedge
Hint: prove that its cauchy
or actually even better - utilize the last question
dense portal
thats not what the first question says
ok, whats x_n - x_n-1 by the recurrence relation
rich river
ok, whats x_n - x_n-1 by the recurrence relation
from the equation?
do u mean xn+1?
what will we do of xn-3 term
rich river
ok, whats x_n - x_n-1 by the recurrence relation
1/2(xn-1 - xn-3)
modern sedge
might be beneficial to just try and see what the sequence does
take x1 = 0 and x2 = 1 and try to compute few terms
rich river
might be beneficial to just try and see what the sequence does
x2 is like the upper bound
all the other numbers are less than x2
modern sedge
yes, that's a good observation
what else do u notice if u start with 0, 1?
what happens to the difference |x_n - x_(n-1)|? (i.e. the distance between conseuctive terms)
rich river
what else do u notice if u start with 0, 1?
it starts decreasing
modern sedge
it starts decreasing
can u just compute the first few terms?
modern sedge
what happens to the difference |x_n - x\_(n-1)|? (i.e. the distance between cons...
Would u be able to make a reasonable guess at what the difference could be?
how does the gap between consecutive terms shrink?
rich river
i.e. r = 1/2 in geometric sequence
yess
modern sedge
oh, sorry if it wasnt clear
modern sedge
Once we actually prove that the gap shrinks by factor of 2 at every step, we will be able to apply this
rich river
oh, sorry if it wasnt clear
no i just missed classes of cauchy
so its like a weak point rn for me
rich river
Once we actually prove that the gap shrinks by factor of 2 at every step, we wil...
lemme try
2
rich river
what happens to the difference |x_n - x\_(n-1)|? (i.e. the distance between cons...
< 2?
ephsilon = 2?
modern sedge
We dont need any epsilon here
you already proved this, now if we are able to prove that |x_(n+1) - xn| < r^n, we can conclude that our sequence is cauchy and so it converges
rich river
you already proved this, now if we are able to prove that |x_(n+1) - xn| < r^n, ...
we cant use this in exam tho :(
modern sedge
Hmm.. so u want a method which doesnt reference the preceeding question?
rich river
Hmm.. so u want a method which doesnt reference the preceeding question?
yess
my professor would actually marks if we used that directly
modern sedge
I see. Well, there is actually a way to write out an explicit formula for this sequence
i dont know how to motivate that solution properly though
spiral remnant
what is the problem in this question?
rich river
wait we proved the geometric series
modern sedge
Well, basically we take a wild guess and say that x_n = r^n
rich river
so can we not do the epsilen
rich river
Well, basically we take a wild guess and say that x_n = r^n
😭 that will count as example
spiral remnant
wait we proved the geometric series
for a geometric series yk if r<1 then sigma is r^n/1-r
modern sedge
this isnt really geometric series though, is it?
rich river
this isnt really geometric series though, is it?
nah
spiral remnant
yk triangle ineuqality?
modern sedge
oh, so they just re-proved the previous result lol
rich river
oh, so they just re-proved the previous result lol
ohhh
modern sedge
This is the important observation that we already made
the rest is just repeating the proof of the last question
rich river
This is the important observation that we already made
yess
can u explain the limit part 😭
modern sedge
the one where we evaluate the limit?
rich river
yes
modern sedge
How did the book do it? Did they define sth like I_n = 2x_n + x_(n-1)?
Or did they find an explicit formula for x_n
rich river
How did the book do it? Did they define sth like I\_n = 2x\_n + x_(n-1)?
why only odd indices
modern sedge
Okay, interesting
rich river
we dont need to follow the book tho
modern sedge
we could notice that $2x_{n}=x_{n-1}+x_{n-2}$
ocean sealBOT
MathIsAlwaysRight
modern sedge
and if we subtract 2x_(n-1), we get
rich river
yes
modern sedge
$2\left(x_{n}-x_{n-1}\right)=x_{n-2}-x_{n-1}$
ocean sealBOT
MathIsAlwaysRight
modern sedge
$\left(x_{n}-x_{n-1}\right)=-\frac{1}{2}\cdot\left(x_{n-1}-x_{n-2}\right)$
ocean sealBOT
MathIsAlwaysRight
rich river
**MathIsAlwaysRight**
wait huh
rich river
thats what u get after subtracting 2x_(n-1)
xn-1 has the xn-3 term tho
from the equation
modern sedge
**MathIsAlwaysRight**
I just did literal subtraction, not applying the eq
$2x_{n}-2x_{n-1}=x_{n-1}+x_{n-2}-2x_{n-1}$
ocean sealBOT
MathIsAlwaysRight
modern sedge
literally like this
dapper herald
I see. Well, there is actually a way to write out an explicit formula for this s...
You can use Characteristic Equation for the recursion.
modern sedge
the point of this is to show that the difference between consecutive terms is actually (-1/2)^n (We already know that the absolute difference is (1/2)^n)
rich river
thats what u get after subtracting 2x_(n-1)
yess
dapper herald
Wait a minute
dapper herald
@modern sedge for Cauchy aren't you supposed to proof for any arbitrary m,n?
not justn+1 and n
modern sedge
Yeah, they did it by that triangle inequality
modern sedge
and @rich river proved it in last exercise (in a more general form), so i didnt consider it necessary to go through it again
modern sedge
<:sippy:920835242980442113>
now that we know the difference, we actually know every term
the sequence starts with 1, 2, then 2 + (-1/2)^1 then (2 + (-1/2)^1) + (-1/2)^2, etc...
so we actually did find the explicit formula, just by knowing the difference between consecutive terms
the sequence is just partial sums of (shifted) geometric series
1, 1 + (-1/2)^0, 1 + (-1/2)^0 + (-1/2)^1, 1 + (-1/2)^0 + (-1/2)^1 + (-1/2)^2, ...
rich river
so we actually did find the explicit formula, just by knowing the difference bet...
yesss
rich river
1, 1 + (-1/2)^0, 1 + (-1/2)^0 + (-1/2)^1, 1 + (-1/2)^0 + (-1/2)^1 + (-1/2)^2, .....
wait how so we get the limit from this
modern sedge
1, 1 + (-1/2)^0, 1 + (-1/2)^0 + (-1/2)^1, 1 + (-1/2)^0 + (-1/2)^1 + (-1/2)^2, .....
and computing the limit of this sequence is just 1 + geometric series with ratio -1/2
rich river
and computing the limit of this sequence is just 1 + geometric series with ratio...
can u do for this question 😭
as an example
modern sedge
can u do for this question 😭
wdym?
modern sedge
1 + (-1/2)^0 + (-1/2)^1 + (-1/2)^2 + ...
well, we just need to compute this
(-1/2)^0 + (-1/2)^1 + (-1/2)^2 + ...
This part is geo-series, with a known formula 1/(1 - r)
in this case, r is -1/2
so we'd get 1/(1 + 1/2) = 1/(3/2) = 2/3
and then 1 + 2/3 = 5/3
thank u so much
modern sedge
Btw there is one more approach for these kind of so-called linear recurrences.
If we guess the solution r^n and we plug it in the equation, we get
2r^(n+2) = r^(n+1) + r^n, which upon division by r^n becomes
2r^2 = r + 1 or
2r^2 - r - 1 = 0. This is known as a "characteristic equation" of the given recurrence
if we solve it, we get r = 1, r = -1/2. This means that
1^n and (-1/2)^n satisfy that recursive formula. And so do all linear combinations of those 2 functions
So the general solution is a*1^n + b*(-1/2)^n = a + b*(-1/2)^n
rich river
im pretty good at recursion
modern sedge
Btw there is one more approach for these kind of so-called linear recurrences.
...
now we could plug in n=1 and n=2 and solve for a, b
rich river
Btw there is one more approach for these kind of so-called linear recurrences.
...
damnn
this is way easier
dapper herald
the sequence is actually a Contractive Sequence
modern sedge
Btw there is one more approach for these kind of so-called linear recurrences.
...
This wasn't strictly necessary here, but it's a pretty cool technique. I wasn't sure whether to do it this way, because i felt like the book wants u to take the cauchy-path
rich river
the sequence is actually a Contractive Sequence
ohhh
modern sedge
using that technique, u could actually figure out an explicit formula for fibonacci sequence btw
rich river
This wasn't strictly necessary here, but it's a pretty cool technique. I wasn't ...
it does want me to take the cauchy path cuz discrete isnt mentioned anywhere :(
dapper herald
If you know about Contractive Sequences then it is pretty easy to identify and then you can proceed accordingly
rich river
If you know about Contractive Sequences then it is pretty easy to identify and t...
i know the definition
modern sedge
Here are some hints on how u could do that
||fibonacci is
f(n) = f(n-1) + f(n-2)
so the characteristic eq would be
x^2 = x + 1
x^2 - x - 1 = 0
which has roots around 1.618, -0.618, so the cauchy formula looks like
a*(1.618)^n + b*(-0.618)^n, where a and b are chosen such that the sequence starts with 0, 1, 1, 2, 3, 5, 8, ...||
dapper herald
Then you don't even need to do anything. just manipulate the given recursion to look like contractive and hence done.
modern sedge
this is how that formula looks like
rich river
this is how that formula looks like
yea this is out of my scope
didnt learn this method
dapper herald
yea this is out of my scope
Nope that is one of the easiest things in whole of math
rich river
Nope that is one of the easiest things in whole of math
not one night before exam 😭
dapper herald
Bro are you Indian?
rich river
Bro are you Indian?
yepp
modern sedge
this is how that formula looks like
its actually pretty simple to derive it, it looks complicated but it only looks this ugly because x^2 - x - 1 = 0 has ugly solutions (1 + sqrt(5)) / 2 and (1 - sqrt(5)) / 2.
rich river
Damn me too
damn which branch bro
rich river
which class??
class??
dapper herald
I am Class 11th tbh
rich river
its actually pretty simple to derive it, it looks complicated but it only looks ...
yesss
rich river
hm?
dapper herald
Do you have more problems like this bro?
rich river
Do you have more problems like this bro?
yeaa
dapper herald
Send it in bro
dapper herald
Yesh
rich river
is this a book?
yess
dapper herald
damn bro have nice day.
modern sedge
is this understanding analysis by abott?
rich river
is this understanding analysis by abott?
mine is robert g bartle
modern sedge
oh interesting
dapper herald
mine is robert g bartle
Bartel and Sherbert?
rich river
Bartel and Sherbert?
yesss
rich river
Good luck on your exams
tysmm
dapper herald
Oh that shii goated asf
modern sedge
i was working with x1 = 1 and x2 = 2, as the answer-key suggested
rich river
that was other question
modern sedge
x1, x1 + (x2 - x1) * (-1/2)^0, x1 + (x2 - x1) * (-1/2)^0 + (x2 - x1) * (-1/2)^1, ...
we can just rescale it basically
the difference will be (x2 - x1) * (-1/2)^n
this makes sure that the first difference is just x2 - x1
rich river
the difference will be (x2 - x1) \* (-1/2)^n
modern sedge
now if u factor that out, you get
x1 + (x2 - x1) * ((-1/2)^0 + (-1/2)^1 + (-1/2)^2 + (-1/2)^3 + ...) =[in the limit]= x1 + (x2-x1) * (2/3)
rich river
now if u factor that out, you get
x1 + (x2 - x1) \* ((-1/2)^0 + (-1/2)^1 + (-1/2...
OHHHH
makes sense
thanksss
modern sedge
np
lone heartBOT
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Available help channel!
lament thistle
help for II.5.b) plssssss
lone heartBOT
help for II.5.b) plssssss
sour verge
,rccw
ocean sealBOT
carmine moss
is it possible to translate to English?
carmine moss
medium of instruction is different
im learning in chinese and english here
idk french
i will try to translate myself hold on
lone heartBOT
@lament thistle Has your question been resolved?
lone heartBOT
@lament thistle Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
gusty fable
hi
gusty fable
ocean sealBOT
gusty fable
My prof didnt fully solve the last question and i was wondering why he cancelled out the work of the weight (W(P))
light jetty
work is 0 when displacement and force are in the same direction i believe
net force
wait hang on im talking bs
lol
did you draw a freebody
gusty fable
did you draw a freebody
uhhh i think
light jetty
where
gusty fable
am sending it rn
light jetty
okay sorry *work is 0 when displacement and force are perpendicular to each other lol
light jetty
i was boutta say 😭
lament thistle
terminale ça non ?
gusty fable
comment terminale
lament thistle
comment terminale
bah c’est des exo de physique terminale ? je me trompe ?
gusty fable
bah c’est des exo de physique terminale ? je me trompe ?
uhh c'est physique de premier annee baccalaureatte
sorry am not that good at french 😭
light jetty
oui oui
light jetty
usually i make them into an "even" freebody
hang pn
like ths
where the red can be derived from Fg
that helps makes things easier
gusty fable
3
light jetty
on question 3?
lowk i have no clue cuz its asking for angular velocity 😭
maybe he was showing a proof
gusty fable
could be
light jetty
unless im mistaken cuz it looks like the work above has the answer
gusty fable
cuz we have the angular velocity for B
light jetty
is there another question he was going over because this looks like work-energy theorem to me
gusty fable
nah that part is for the last question
light jetty
okay well he cancelled it because the work there is 0
gusty fable
is there another question he was going over because this looks like work-energy ...
yeah it is that
gusty fable
okay well he cancelled it because the work there is 0
yeah but am wondering why its 0
light jetty
and its for the pulley?
gusty fable
am fairly certain that is the net work for the pulling mechanism
light jetty
all 3 together make the net yeah
what does the W(R) and W(F) stand for
thats whats messing me up because we use different variables here lol
gusty fable
the R is for the plane reaction (litteral translation)
F is the force the solid S applies on the pulley
light jetty
im assuming plane reaction is normal force maybe
gusty fable
its like friction
gusty fable
in this case it doesnt matter cuz its negligeble
light jetty
is this for the pulley
light jetty
okay well if its for the actual pulley
the pulley isnt moving up or down, so theres no displacement
if there's no displacement then is 0
light jetty
nah its okay, it took me awhile to understand the question lol
gusty fable
thought the force weight would cancel out cuz of cos 90 or sum😭
light jetty
that could also be the wrong explanation but it works
light jetty
thought the force weight would cancel out cuz of cos 90 or sum😭
that is also true in general of work
gusty fable
THANKSSS
gusty fable
but its not moving, its spinning but stationary
yeah
light jetty
stationary objects have no work
gusty fable
Fg isnt bring it down either
wait whats Fg
gusty fable
ohhh okok
sinful saffron
How is asymptotic with respect to n specified?
lone heartBOT
How is asymptotic with respect to *n* specified?
sinful saffron
I have the equation* $\ln\left(\frac{d}{dz}P^{n}\left(z\right)\right)\sim\ln\left(p-p^{z}\right)\ln\left(P^{n}\left(z\right)\right)$
*idk what things like these are called when its no longer equalities or inequalities or a single expression
and want to specify it asymtoptic with repsect to $n$
vale wigeon
i mean, if the variable is called "n", then chances are it's implied to approach +∞
ocean sealBOT
Aper
vale wigeon
you can also just say it in words
ocean sealBOT
Aper
vale wigeon
or like... shove an $n \to +\infty$ under the tilde? but that's gonna get ugly
ocean sealBOT
Ann
sinful saffron
actually I could just do n under the tilde, the assumption would for asymptotes is that its going to infinity anyway
I'll do that
sinful saffron
or like... shove an $n \to +\infty$ under the tilde? but that's gonna get ugly
ty
.close
alpine sable
thats the formula
manic quail
yep, those ones. Try applying the second one
alpine sable
hmm okay
alpine sable
yep, those ones. Try applying the second one
i dont really know how😅
manic quail
wait, first, do you want to solve 11 or 12?
alpine sable
i want to do 11
manic quail
you choose some value for a and some for b and see if it works out
alpine sable
hmmm
manic quail
(3x+5y)² seems like a great choice to me xD
neat ruin
Uhhhm can any one tell me why