Yep, perfect.
#help-36
glossy zephyr
I hope you can see that this inequality can be described into two "parts", y = 30x / (x+30), which is a curve and y < 30x / (x+30), which is the area "under" the curve.
We will use y = 30x / (x+30) as a function to find what numbers will be included into S
We start constructing from the empty set S = {}
Our first candidate for an element will be 1, since it essentially will minimize xy.
S = {1}
next we want to find the next integer x that result as part of the function is bigger that all elements "y" in S.
we will try next with 2
y = 30 * 2 / (2+30) = 60 / 32 = 1.875
1.875 is bigger than all elements "y" from S, so we will add it.
S = {1,2}
And you keep on doing that until you fill all elements
Here knowing about a bit of calculus is helpful, since it tells you that you cant have more than 2 elements bigger than 30.
arctic sand
>And you keep on doing that until you fill all elements
Here knowing about a bit...
can i still do it if i dont know anything abt calculus?
glossy zephyr
Yep, theres a fast method for it.
Take your largest y, plug it in the place of y in the function, and isolate x.
Ill showcase for the next element.
S={1,2}. the biggest element is 2,
2(30+x) = 30x
60 + 2x = 30x
60 = 28x
60/28 = x = ~2.14```Since we are interested in integers, youll have to round up.
So our next element is 3.
You end up with S={1,2,3}
Youll have to repeat a few times, and youll eventually see that youll get a number x = a negative, at that point, you already found S. If you really want, go ahead and try to construct the full S.
plain rain
,tex
Hello there. So I have a question about what the following exercise means when it says they "all converge". Does it mean they converge to the same number, or is it not specified?
\
$ a_{pn} , a_{pn-1} , \dots , a_{pn-(p-1)} $ and $a_{qn} $ , with GCD$\left( q, p\right) = 1 $ all converge, then $ a_n $ converges
final saddleBOT
,tex
Hello there. So I have a question about what the following exercise means w...
soft zealotBOT
fijokazż
soft zealotBOT
plain rain
its the one under 1.9
says: if the subsequences ... of the sequence a_n converge, where p,q coprime, then prove that a_n converges too
the author def shouldve specified bc its the first exercise on this kinda stuff
rugged merlin
,tex
Hello there. So I have a question about what the following exercise means w...
I dont think it means they converge to the same number
Rather all subsequences have valid limits
rugged merlin
plain rain
its prolly a tough proof for my standards so i dont wanna start with the wrong facts
rugged merlin
Think of it this way
If they all converged to thr same number
The problem becomes like
Trivial 
plain rain
wed not need a_qn right?
rugged merlin
And its probably not meant to be trivial
plain rain
Trivial <:kekhands:1294941772841091112>
hell naw man i wouldnt know how to that either
rugged merlin
Yea
plain rain
do*
plain rain
2n and 2n-1 right?
rugged merlin
If every "slice" of the sequence converged to the same number L then obviously the entire sequence itself converges to L
plain rain
broski its easy to just say it, but try proving that shit
altho i do believe its an easier problem
ill take ur word for it
it makes sense to not be the same number
not necessarily at least
thanks!!
.close
final saddleBOT
rugged merlin
broski its easy to just say it, but try proving that shit
I mean you can apply the epsilon delta definition on each limit
rugged merlin
Then you use the division algorithm
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gleaming halo
two circles S1 = 0, S2 = 0 of equal radius 'r' intersect such that the center of one circle passes through the center of other circle. Another circle (S=0) touches S1 = 0 internally and S2 = 0 externally and also touches the line joining the centers of two circles S1= 0 and S2 = 0. Then the radius of S = 0 is?
final saddleBOT
two circles S1 = 0, S2 = 0 of equal radius 'r' intersect such that the center of...
gleaming halo
This is all I have tried.
lucid nymph
and you arent given any distances?
wouldnt that mean you should just find it in terms of r?
final saddleBOT
@gleaming halo Has your question been resolved?
gleaming halo
wouldnt that mean you should just find it in terms of r?
yes
storm karma
@gleaming halo maybe just try putting the value of x in 2nd eqn and just simplify
gleaming halo
<@948924905154543656> maybe just try putting the value of x in 2nd eqn and just ...
I dont think it can be simplified...
or it will just take stupidly long time
tranquil pine
hi, i need to find the inflection points of this function.
i get (2, -36) as a point but the answers list (4/3, -398/27), im not sure where i went wrong
final saddleBOT
hi, i need to find the inflection points of this function.
i get (2, -36) as a p...
tired walrus
show your work
also show the answer key maybe
cause the answer they gave you seems kinda sus
tranquil pine
im on a laptop and i did it on a piece of paper
pg 205 #3a
tired walrus
that's odd
my answer matches yours and the book's answer is just out-the-window wrong lmao
even with desmos, f''(2) = 0 for the function in 2a
does the book have any errata
left echo
,w inflection point of x^3 - 6x^2 - 15x + 10
soft zealotBOT
tranquil pine
errata?
left echo
well there you go
tired walrus
errata?
a thing at either the beginning or end to list known typos and fixes in the book
tranquil pine
a thing at either the beginning or end to list known typos and fixes in the book
oh, no it doesnt
well at least i got the right answer
tired walrus
are you under a legal requirement to comply with the book explicitly above mathematical truth
final saddleBOT
@tranquil pine Has your question been resolved?
vapid haven
is the ans 2?
final saddleBOT
tiny kraken
<@&268886789983436800>
covert lodge
how is this possible
final saddleBOT
how is this possible
covert lodge
here i first took x as tan theta
so after a lil simplifying i get the final value to be 4 theta
tired walrus
,w simplify (1-tan^2(x))/(1+tan^2(x))
soft zealotBOT
tired walrus
so you ended up with cos^-1(cos(2θ))
covert lodge
yes
tired walrus
and let me guess, you simplified that as just 2θ?
tired walrus
you declared θ := tan^-1(x) thus θ ∈ (-π/2, +π/2) thus 2θ ∈ (-π, π)
tired walrus
if 0 ≤ 2θ < π then yes cos^-1(cos(2θ)) = 2θ and the expression is equal to 4θ
but if -π < 2θ < 0 then cos^-1(cos(2θ)) = -2θ instead
covert lodge
😮
shoot
so then
if 2 theta lies between -pi to 0
so theta = tan inv x
now what
tired walrus
θ ∈ (-π/2, 0] so x = tan(θ) ∈ ?
covert lodge
-pi/2 to 0
ok so
im getting -inf,0
the inequality does change if i can all sides right
im talking abt -pi/2 < tan^-1 x < 0
mb it wont
cuz tan^-1 is a inc function
what its a
oh silly mistake mb
tysm i got it
tiny kraken
<@&268886789983436800>
covert lodge
,close
vital crag
it's .close
covert lodge
.close
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dreamy tartan
my teacher provided this example to explain small o notation
final saddleBOT
my teacher provided this example to explain small o notation
dreamy tartan
but then he asks me to prove this
i'm not really sure where he got the x in "x + o(x)" in the example so idk how to arrive at "1-x^2/2" here
tired walrus
f = o(g) means lim f/g = 0
in particular o(1) denotes any function which approaches 0
what does "per casa" mean btw
dreamy tartan
"for home"
tired walrus
ok
your teacher's example shows sin(x) - x = o(x)
add the x to both sides and you get sin(x) = x + o(x)
dreamy tartan
right
tired walrus
in a similar vein, first rewrite lim[x->0] (1-cos(x))/x^2 = 1/2 as a limit of (something) -> 0
and then get that (something) as a fraction
dreamy tartan
AH! i see now
tired walrus
and you'll have numerator = o(x^2)
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worthy musk
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worthy musk
I need help on the third question
Idk if I change it or what
<@&286206848099549185>
crisp mason
do yk the summ difference formulas/equations?
worthy musk
You see for the first question I changed sine to cos and stuff
worthy musk
do yk the summ difference formulas/equations?
nope
worthy musk
dis all I learned
so far
and the first pic I sent is what I am learning now in trigonometry
Did
crisp mason
can you not
wooden heron
Bro sin 345 is sin 15 am I wrong
worthy musk
is it supposed to change tho
wooden heron
U just add a negative sign
worthy musk
like how does it bcome the opposite
wooden heron
Use coterminal angles
crisp mason
Bro sin 345 is sin 15 am I wrong
yes
worthy musk
Its sin 75
desert delta
instead of thinking of 345 as 270+75, think of it as 360-15
crisp mason
sin isnt positive in the 4th quadrant
wooden heron
Dude I know
That’s why I said add a negative
The value is the same
It’s just negative
worthy musk
sin isnt positive in the 4th quadrant
there’s sin in the 4th quadrant?
wooden heron
U don’t need to do sin(alph plus beta)
wooden heron
there’s sin in the 4th quadrant?
??
worthy musk
I thought it’s js cos and sec
wooden heron
yeah
worthy musk
worthy musk
instead of thinking of 345 as 270+75, think of it as 360-15
actually?
wooden heron
Oh sh im wrong
Yeah u do use the addition property
Mb
Do sin(360-15)
Which is js sin360cos15 - sin15cos360
sin15 is a half angle which means its root of 1-cos30/2, or root of 2-root3/2
wooden heron
No
crisp mason
there’s sin in the 4th quadrant?
theres sin in all quadrants my friend, its only positive in the first and second
worthy musk
worthy musk
theres sin in all quadrants my friend, its only positive in the first and second
so what’s dis graph
wooden heron
Bro those papers do not do you justice
worthy musk
I’m js saying bro I started trigonometry two days ago
wooden heron
So did I lol
crisp mason
to make your life easier why not write $csc(x)$ instead of $cosec(x)$
soft zealotBOT
Cowking
worthy musk
What’s the x
?
oh that x is for undefined my teacher said
<@&286206848099549185>
crisp mason
oh that x is for undefined my teacher said
the x is a variable, it just varies in use depending on what youre talking about and how its being used
crisp mason
<@&286206848099549185>
chill
final saddleBOT
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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
wooden heron
Were u following what I was saying
worthy musk
Were u following what I was saying
I don’t understand what You said
crisp mason
so what do I do
try sum and difference formulas, if you really dont
wooden heron
What is sin(360-15)
wooden heron
.
Use ur subtraction property of sine
This is not algebra
Friend me rain I will show you
wooden heron
I dont have a whiteboard
final saddleBOT
@worthy musk Has your question been resolved?
final saddleBOT
@worthy musk Has your question been resolved?
fickle junco
between 2 touching circles and a common tangent to both, can you always draw a smaller circle external to and touching all three? if so, what is the proof?
final saddleBOT
between 2 touching circles and a common tangent to both, can you always draw a s...
fickle junco
it's an intermediate step which if true would solve a geometry problem I was doing, but idk how to prove it
lucid nymph
coordinate geometry would probably suffice
fickle junco
its a setup like this
fickle junco
coordinate geometry would probably suffice
yeah hopefully, but i was looking for a nicer proof bc i feel like co-ordinate geometry would take forever and be messy
silent shore
I seen that before
fickle junco
Not sure if you can do anything with that though
maybe let the third external circle be infinite radius works i think
which turns it into a line
lucid nymph
yeah hopefully, but i was looking for a nicer proof bc i feel like co-ordinate g...
you can find an equation for the radius of the smallest circle in terms of the other radii and as long as there are no values for the two circles that give a negative or imaginary result then this should be proven
sacred locust
maybe let the third external circle be infinite radius works i think
Yes I think that could definitely work
Worth a try
fickle junco
you can find an equation for the radius of the smallest circle in terms of the o...
aren't you assuming it is possible then showing it works for all configurations where its possible, not that its possible for all configurations?
final saddleBOT
@fickle junco Has your question been resolved?
whole halo
its a setup like this
search up ford circles for a demonstration of this happening
the radius of that middle circle is given as:
fickle junco
aren't you assuming it is possible then showing it works for all configurations ...
Uhhh so I've thought abt this, is this fine bc in co-ordinate geometry relations like this are equivalent, and not one-way implications?
fickle junco
the radius of that middle circle is given as:
yeah proving that was part of the problem lolol
whole halo
consider rotating a tangent line around one of the circles
whole halo
its a setup like this
it only then becomes tangent to the other circle at two places which both match the diagram
coordinate geometry can work to prove this
I will note there is a third option
fickle junco
aren't you assuming it is possible then showing it works for all configurations ...
this was the problem which i had with using co-ordinate geometry
whole halo
coordinate does not have a - in it
if you draw the tangent line right between the two circles, the third circle cannot be drawn
that counterexample should be enough to answer the question as a "no"
you can pore over the problem to see if this case is allowed, if so then thats the only kind of way that the 3rd circle cannot exist
in any other rotation, the line either crosses the circle twice or none at all (which dont count for tangent), or it crosses it once matching the diagram
fickle junco
if you draw the tangent line right between the two circles, the third circle can...
lol fair, this definitely isnt relevant to the problem though, I shouldve clarified at the start
whole halo
you didnt clarify much of anything
then again theres no way you can anticipate someone else's questions
whole halo
np
coordinates can be tedious, you can WLOG a lot of the necessary details out
use discriminant = 0 as usual to determine tangency
fickle junco
.close
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mossy locust
in a game of three player nim, with stacks of 2 and 1, what move should the first player do?
final saddleBOT
in a game of three player nim, with stacks of 2 and 1, what move should the firs...
bleak granite
what's the goal of "three-player nim"?
mossy locust
to be the last player to lose.
viral jungle
I suppose then by that you mean be the player to grab the last object
mossy locust
well yes
loud sundial
what's the goal of "three-player nim"?
hi 
bleak granite
in most of those games I think the 2nd player always wins? they just have to make groups of 3, right? 
if I understand correctly
bleak granite
hi <:happy:908558103992225822>
opal plinth
There is no winning strategy (unless the strategy of the other players is known)
mossy locust
but if there is no way for player x to be the last player to take the last object, they should idealy try to be the second to lose rather than the last
mossy locust
There is no winning strategy (unless the strategy of the other players is known)
why exactly?
mossy locust
in most of those games I think the 2nd player always wins? they just have to mak...
not really?
like in any stacks of 1 where there are an even number of stacks
viral jungle
define "second to lose"
mossy locust
define "second to lose"
once one player loses, the other two players continue playing until someone loses again
bleak granite
not really?
yeah I just looked it up, I definitely do not understand this game, so I will let someone better equipped to answer this question, my apologies 🙇♂️
mossy locust
its ok
viral jungle
wait so there are two piles?
opal plinth
why exactly?
From the point of view of one player, the other two players can form a team at any point, and a 2 vs 1 is almost always guaranteed to end in a loss for the 1
viral jungle
you might want to clarify the rules of this new variant then
mossy locust
wait so there are two piles?
yes, one of size 2 and one of size 1
mossy locust
you might want to clarify the rules of this new variant then
oh sorry
ill do that
the three players play in a loop removing any amount of items from any one stack, once someone cannot make a move, they lose, then the other two players continue in that same order until someone cannot move again
because a player can only not move if there is nothing left, once one player loses, the next player will also imidiatly lose
but they lose "less" because they lost second
opal plinth
Hold on do you really mean that the game starts with only 2 stacks, one of size 2 and one of size 1? I understood your question as a general game of n stacks, just with the restriction that stacks can only be of size 2 or 1...
mossy locust
Hold on do you really mean that the game starts with only 2 stacks, one of size ...
oh sorry
im looking at this specicif game of size 1 and 2
because then that gets much more complicated
its just that 2+1 is the smallest amount of stacks where there isnt one obviouse winning soluition for a player
like in any singular stack, the first player takes the whole stack, the second player loses, and the third player 'draws', leaving the first player as the winner
then a similar thing can be seen for 1+1 but its the second player who wins, third player who loses, and first player who draws
opal plinth
So the player who takes the last item wins, the next loses, and the next draws?
opal plinth
And your question is about the first player's move so that they win or, if that's not possible, draw?
opal plinth
Right, well it's just going to be "take the stack of 2"
The next player wins by taking the last item, the next loses
So the first player draws
The other two possible moves are "take the stack of 1" which can lead to a loss or a draw, and "take one from the stack of 2", which definitely leads to a loss
mossy locust
mh yeah
alright thanks
i was just being silly
i wonder if there are any cases in which they would wanna take the stack of 1
aight
.close
final saddleBOT
opal plinth
To make the second player win
That implies the first and second players are cooperating in some way
mossy locust
That implies the first and second players are cooperating in some way
but when would that happen?
(should i reopen it?)
opal plinth
Idk, who is playing?
viral jungle
.reopen
final saddleBOT
✅ Original question: #help-36 message
opal plinth
.reopen
final saddleBOT
✅ Original question: #help-36 message
opal plinth
Weird bot
mossy locust
Idk, who is playing?
wdym?
opal plinth
You're asking "when would that happen" to "the first and second players are cooperating"
Kind of already answered, no?
mossy locust
i mean when would that happen, like when would they cooperate insted of just going for themself
opal plinth
mossy locust
sorry im kinda not great at this
opal plinth
I'm talking about meta cooperation
final tangle
friends, don't like the third person
opal plinth
It's not momentary cooperation within the game from two players just to serve their own interests better
It's cooperation "outside" the game, to make the last player lose
mossy locust
so like if they are playing this game and then another game in which if player 3 wins in this game they will win in the second game?
opal plinth
No you're overthinking this
It's a three player game, so two players could just be "friends" like ramonov just said
opal plinth
If the first player takes the stack of 2, they guarantee they will only draw
But that means the second player loses
mossy locust
But that means the second player loses
but how?
the second player goes, takes the 1 stack, then the third player loses
opal plinth
Uh hold on I got it mixed up
final tangle
if they take 1 from the stack of 2,
P3 wins
P1 loses
p2 draws
opal plinth
Yeah sorry this needs to be even more specific
If players 1 and 3 are in a team, they don't want player 2 to win
Only option is to take just one item from the stack of 2
The question now is what value is placed on winning, drawing, and losing
final tangle
P3 has no control here
opal plinth
True
mossy locust
The question now is what value is placed on winning, drawing, and losing
i feel like what could make sense is +1 for winning, 0 for drawing, and -1 for losing
opal plinth
But P1 can make P3 win
Let's just list the options
Option A: taking the stack of 2, then P1 draws, P2 wins, P3 loses
Option B: taking 1 from the stack of 2, then P1 loses, P2 draws, P3 wins
Option C1: taking the stack of 1, then P2 takes all, then P1 draws, P2 wins, P3 loses (same as option A)
Option C2: taking the stack of 1, then P2 takes only 1 item, then P1 loses, P2 draws, P3 wins (same as option B)
Do we agree?
mossy locust
yes
opal plinth
i feel like what could make sense is +1 for winning, 0 for drawing, and -1 for l...
Then if we follow these values
If P1 and P3 are in a team, picking option B if optimal for the team
But option A is optimal for P1 if they aren't in a team (or don't care)
Basically preventing P2 from winning
mossy locust
or if P1 and P2 are in a team
so depending on the team, P1 should choose option A or B
opal plinth
If P1 and P2 are in a team then P1 goes for option A anyway, it doesn't matter
mossy locust
yeah thats what i mean
opal plinth
Well the only team that matters is P1 and P3
mossy locust
why wouldnt P1 and P2s team be valid? beucase if they are on a team P1 should choose A
opal plinth
Yes it doesn't change what P1 would do
Or let's say you're doing many games of this (not necessarily the same starts, or maybe this stack of 1 and stack of 2 situation is the end of the latest game), you're keeping score, and P2 has say 9 points while P3 has only 6
You win the series when you get to 10 points
Then P1 would not want to let P2 win the series
Even if they lose a point, they keep the series going by choosing option B
opal plinth
So yeah, 3-player game theory is complicated
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vague slate
I need help understanding how pi can and can't be used
final saddleBOT
I need help understanding how pi can and can't be used
worthy wren
that is quite vague. any field u r interested in? π shows up a lot in various fields
vague slate
Primarily geometry and number theory
Is what I really need help understanding pi with
vital crag
Primarily geometry and number theory
Still too vague. What are you actually learning that involves pi
vague slate
I forgot exactly what I was wanting for number theory but how It can help calculate shapes like circles and other related
Shapes
vital crag
Calculate what about shapes
vague slate
Circumface and area as well volume
crimson zenith
I need help understanding how pi can and can't be used
vague slate
[this might be good for a start](https://en.wikipedia.org/wiki/List_of_formulae_...
I'll start reading it
glossy zephyr
Circumface and area as well volume
π is a ratio people have known about (and found approximations) for a really long time.
If you take a circle's perimeter and divide it by its diameter as two lengths; then the result is always equal to π
even beyond the obvious cases where you see a circle, or some similar shapes, it usually appears all around trigonometry and any other case where rotational movement is involved in some way or another.
vague slate
even beyond the obvious cases where you see a circle, or some similar shapes, it...
Okay
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warped burrow
could anyoen sketch out this problem? or explain what is meant by 18 degrees to the horizontal? I can't picture the layout of this problem in my head
final saddleBOT
could anyoen sketch out this problem? or explain what is meant by 18 degrees to ...
cold rover
could anyoen sketch out this problem? or explain what is meant by 18 degrees to ...
Hi, let me explain how you can draw this image urself
First off, let's use some common sense: how does a ship move relative to its front?
warped burrow
forward?
cold rover
Yep, pretty simple
Now, we have the bird referenced with a horizontal
So I'll draw a diagram...
cold rover
Now, the bridge flies 18° to the horizontal (dotted line!) from front of ship to back of ship. How will it fly in this diagram?
cold rover
Yep 👍
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dense garnet
Suppose $y=Ax$ where $y$ is $p\times 1$, $A$ is $p\times n$ and $x$ is $n\times 1$. Suppose further that there are exactly $m\leq p$ linear, homogeneous equations in the components of $y$, i.e. such that each equation equals $0$. Can we say anything about the rank of $A$?
final saddleBOT
Suppose $y=Ax$ where $y$ is $p\times 1$, $A$ is $p\times n$ and $x$ is $n\times ...
soft zealotBOT
psie
dense garnet
.close
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warm python
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warm python
I'm assuming this means using the definition which I will attach now
nvm, got it
I use partiton of length k/n and n goes to infty , which I can by this theorm
abstract mango
all set ?
warm python
I think so, I just need to compute the lower and upper sum using this and show they're equal no?
abstract mango
yep.
warm python
I know x^2 is continuous so it is integrable ( which is a fairly important detail tbf)
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fossil kiln
Is this right? If so, how do i simplify it the answer does not contain any trigonometric ratio.
-# excuse my awful ahh writing
🤦♂️
.close
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robust sequoia
I know this is not about maths but I just want to know which direction the arrows for U1 and U2 should be and which curve represent which tension
final saddleBOT
I know this is not about maths but I just want to know which direction the arrow...
soft zealotBOT
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@robust sequoia Has your question been resolved?
blazing nymph
Hi
blazing nymph
tried using the difference of squares right
but unsure where to go from there
like i ave |xn - x| < (sqrt(xn) + sqrt(x))*eps
scarlet sequoia
tried using the difference of squares right
Depends on what x is
blazing nymph
xn >= 0 for all n btw
scarlet sequoia
if x is 0, you can't use difference of squares
blazing nymph
so consider that case seperately
scarlet sequoia
like i ave |xn - x| < (sqrt(xn) + sqrt(x))*eps
Isn't that the other way around that you'd like it?
you need |sqrt(x_n) - sqrt(x)| < ...
or are you trying to find delta that way
blazing nymph
you need |sqrt(x_n) - sqrt(x)| < ...
yea so i want to show that this is less than eps, just manipulated this
scarlet sequoia
ok
subtle wave
I think maybe try to prove$|\sqrt{x}-\sqrt{y}| \le \sqrt{|x-y|}$
soft zealotBOT
vSong
scarlet sequoia
like i ave |xn - x| < (sqrt(xn) + sqrt(x))*eps
so you want the RHS to be bigger than delta basically
scarlet sequoia
I think maybe try to prove$|\sqrt{x}-\sqrt{y}| \le \sqrt{|x-y|}$
Uh....
my bad I got the order wrong
scarlet sequoia
we have to introduce delta?
so if |xn - x| < delta
basically you want delta < (sqrt(xn) + sqrt(x))*eps
so that |xn - x| < that value
blazing nymph
scarlet sequoia
you need |sqrt(x_n) - sqrt(x)| < ...
and so |sqrt(x_n) - sqrt(x)| < eps
onyx peak
x = 4, y = 1
(it's true btw, and provable)
scarlet sequoia
(it's true btw, and provable)
oh wait yeah I got the order wrong
blazing nymph
basically you want delta < (sqrt(xn) + sqrt(x))*eps
Sorry I don’t get this
scarlet sequoia
Sorry I don’t get this
you want x_n close enough to x so that |x_n - x| < (sqrt(xn) + sqrt(x))*eps
But you can't really control that written that way
scarlet sequoia
because the RHS also depends on n
but, if you can prove there's an arbitrary fixed value delta > 0
such that delta < (sqrt(xn) + sqrt(x))*eps for all n
then you win, and you can use the limit definition of x_n -> x
"For all $\delta > 0$, there exists $N\in \bN$ such that $\forall n\geq N$, $|x_n-x| < \delta$"
blazing nymph
Ohhhh I seeee
soft zealotBOT
Rafilouyear2026
scarlet sequoia
Ok, take delta = (that value)
then for all n >= N, |sqrt(x_n)-sqrt(x)| < epsilon
blazing nymph
Yh lemme try
scarlet sequoia
Uh....
Btw I apologize for reading or understanding that wrong, my mind is foggy today
final saddleBOT
@blazing nymph Has your question been resolved?
scarlet sequoia
i let the delta be sqrt(x) * eps
nice, so that delta works
do you see why we had to make a special case for x = 0 in that proof
blazing nymph
division by 0?
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severe canyon
<@&268886789983436800>
old quarry
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old quarry
i drew the diagram
and the radius = half of shortest distance between the 2 curves right
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@old quarry Has your question been resolved?
old quarry
after this idk what to do
main locust
Do you understand derivatives?
yep it can be solved through derivatives and point to line distance formula
old quarry
Do you understand derivatives?
yes
opal plinth
Do you see how the blue segment is the shortest distance between the curves?
old quarry
yes
opal plinth
And it's perpendicular to the green lines, which are tangents
old quarry
yes
opal plinth
So what would their slope be?
opal plinth
What is the slope of the blue segment?
old quarry
i dont know
opal plinth
From symmetry it should be obvious
old quarry
how to find that
opal plinth
Do you see that the curves are symmetric images of each other about some line? What line?
old quarry
Do you see that the curves are symmetric images of each other about some line? W...
huhh y=x i guess? but im not sure
opal plinth
Yes
old quarry
how do u know its symm about that line
old quarry
i mean yeah i know how to get inverse of a function
opal plinth
Well, surely you can see that x = y^2+2 is the inverse of y = x^2+2
old quarry
ohh shit yeah
opal plinth
So they are symmetric about y=x
So can you figure out the slope of the blue segment now?
old quarry
yeah..i can take it from here
old quarry
Well, surely you can see that x = y^2+2 is the inverse of y = x^2+2
tbh its tough to notice such things in exam please dont use words such as surely 😭😭
thank you
.close
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opal plinth
I figured you would have seen it from drawing the diagram
old quarry
i mean i did but i thought i just drew it in a symmetrical way
cause if it had been y=x^2+3 i wouldve drawn in the same way
opal plinth
Oh well, in the future you should pay attention to symmetry, it's often very useful to simplify problems
old quarry
ty
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stable viper
can someone teach me trapezoidal estimate? i have this formula but not sure how to use.
final saddleBOT
can someone teach me trapezoidal estimate? i have this formula but not sure how ...
opal plinth
Imagine a trapezoid with vertices (0, 0), (1, 0), (1, f(1)), and (0, f(0))
Here f(0) = 1 and f(1) = 0.5
What's the area of this trapezoid?
@stable viper
Good talk
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plain rain
,tex
Hello, so i am trying to solve a problem from a textbook, that has the following condition: \
$ a_{pn} , a_{pn-1} , \dots , a_{pn-(p-1)} , a_{qn} $ with gcd$\left( p,q\right) = 1$ converge
\
And I wanna prove that if thats true then $a_n$ converges too. Please don't give me any hints except for answering my following question: Is it in the right path to take the cases of $p>q $ and $ p<q$ ?
final saddleBOT
,tex
Hello, so i am trying to solve a problem from a textbook, that has the foll...
soft zealotBOT
fijokazż
bold zenith
So the indices go: pn, pn-1, ..., pn-(p-1), qn?
plain rain
they are subsequences of a_n
bold zenith
How can a finite sequence converge
plain rain
but theyre not finite, its like having the subsequence a_2n of a_n
bold zenith
Oh like 2,1,q,4,3,2q,6,5,3q,...? for p=2
plain rain
huh
nono
if p were 2 we'd have a_2n and a_(2n-1)
and then a_qn
and wed know that they all converge, not necessarily to the same number afawk
bold zenith
Oh like 2,1,q,4,3,2q,6,5,3q,...? for p=2
Like this?
plain rain
Oh like 2,1,q,4,3,2q,6,5,3q,...? for p=2
i dont rlly get what these numbers are lol
bold zenith
yeah
plain rain
they would be a_2n , a_(2n-1) , a_qn and theyd be subsequences of a_n
so theyd have individual indices
the first would have a_2, 4, 6,...
second is all odds and third is all the multiples of q
bold zenith
If those converge can't you make subsequences by removing the terms with qn and just focus on pn-i
bold zenith
Oh I see
plain rain
i did prove it for them converging on the same one tho
bold zenith
So that's what qn does is connect them all
bold zenith
You'd want to use that a_pn,a_pn-1,...,a_pn-(p-1) has the same limit as a_q,a_2q,a_3q,...
Don't know if q<p matters
Oh you can choose whether q<p or not
Yeah let q<p
Don't need to split cases
plain rain
You'd want to use that `a_pn,a_pn-1,...,a_pn-(p-1)` has the same limit as `a_q,a...
i dont see how to prove this tho
bold zenith
If a subsequence is infinite then its later terms will be fairly close to the limit of the original sequence
Because it can only go away from it so much at that point
plain rain
if i could just say that we wouldnt rlly need a_qn tho no?
i think the goal for the exercise is to prove they converge to the same limit
bold zenith
Oh
plain rain
somehow a_qn ties in
plain rain
its a fucked up problem
bold zenith
If a subsequence is infinite then its later terms will be fairly close to the li...
What's wrong with this vague argument though
plain rain
If a subsequence is infinite then its later terms will be fairly close to the li...
it doesnt convince me
bold zenith
Suppose the limit of the subsequence is different then find the difference between the limits and just let epsilon be smaller than that, and N large enough for epsilon
plain rain
hmm okay ill try that out
ill be back some other day cuz i dont wanna overload myself. i like working slow
thank you for the suggestion, i think itll lead to somewhere
bold zenith
I feel like i might be wrong though
grim nebula
plain rain
I feel like i might be wrong though
its okay if you are. ive spent countless hours pondering on it
bold zenith
If the mathematics works then 
, ignore my worrying, but double check anyway
plain rain
hi snow
bold zenith
You get to choose which conditions you want to use
plain rain
ik that there exist x,y s.t. px+qy = 1
plain rain
You get to choose which conditions you want to use
we dont lol
grim nebula
yeah so basically that fact is gonna be important
bold zenith
If gcd(p,q)=1 then this holds
plain rain
i cant tie that in the game at all tho ive tried so many things
the fact that x,y are completely random makes me so mad
bold zenith
So if we say "if gcd(p,q)=1 and p>q, then this holds" thats okay, but it means we have less to prove the a_n convergence with
But maybe thats still enough, and you can use that p>q for each of the convegences you know to your advantage
grim nebula
the main thing you should consider is how the set {0, q, 2q, 3q, ..., (p - 1)q} is "spread" across the sequences (a_pn), (a_(pn - 1)), ..., (a_(pn - (p - 1)))
plain rain
hmm bbmaths mentioned something similar earlier
alright ill work with that
thanks both of you
.close
final saddleBOT
grim nebula
you can try rearranging the px + qy = 1 equation to get your multiples of q
plain rain
.reopen
final saddleBOT
✅ Original question: #help-36 message
grim nebula
e.g., yq = 1 - px, 2yq = 2 - 2px, 3yq = 3 - 3px etc
plain rain
yeah but y cant be anything we want it to be
plain rain
hm
grim nebula
if px + qy = 1, then p(x - q) + q(y + p) = 1 as well
so any pair (x, y) which solves the equation gives another pair (x - q, y + p) also solving the equation
iterating, you can force y to be positive
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grim nebula
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stable viper
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stable viper
what did i do wrong?
loud sundial
stable viper
oh true
final saddleBOT
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stable viper
wait f(x) i end up getting x^3 -7x^2 + 6x, so when i sub in 1, i get 0
loud sundial
,w expand (x-1)^2 (x-5)
soft zealotBOT
loud sundial
wait f(x) i end up getting x^3 -7x^2 + 6x, so when i sub in 1, i get 0
recheck that
final saddleBOT
loud sundial
ok thanks tysm
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robust sequoia
can someone explain to me how to solve the last differential equation. it's in french
final saddleBOT
can someone explain to me how to solve the last differential equation. it's in f...
@robust sequoia Has your question been resolved?
robust sequoia
<@&286206848099549185>
leaden moon
<@&286206848099549185>
Provide a translation please?
dull sphinx
English please
leaden moon
<@&286206848099549185>
It's not like we'd understand french quicky
robust sequoia
Provide a translation please?
yeah sure
weird but I hope u understand
différential equation verified by i1
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@robust sequoia Has your question been resolved?
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fossil kiln
Winnie likes positive integers whose only prime factors are $7$ and $13.$ So, Winnie likes the numbers $1, 13, 49,$ and $91,$ but she does not like $14, 22,$ or $182.$
What is the sum of the reciprocals of all the numbers that Winnie likes?
final saddleBOT
Winnie likes positive integers whose only prime factors are $7$ and $13.$ So, Wi...
soft zealotBOT
ch3rry
fossil kiln
I think we should be able to reduce it to a converging series or smth idk
worldly mesa
yes
worldly mesa
think what is the prime factorization of a number Winnie likes
fringe light
is the answer 91/72?
fossil kiln
,,\sum_{x=1}^{\infty} \frac{1}{7^x} \times \sum_{y=1}^{\infty} \frac{1}{13^y}
soft zealotBOT
ch3rry
fringe light
yes
fossil kiln
is the answer 91/72?
I dont hv the answer key
fringe light
oh
fringe light
**ch3rry**
this is how i did it
fringe light
yes
fossil kiln
Also we cant strt frm 0 ryt winnie wouldn't like tht
fringe light
you will get 91/72 ig
fringe light
Also we cant strt frm 0 ryt winnie wouldn't like tht
thts why we start frm 1
fossil kiln
thts why we start frm 1
Yh ima kill winnie
fringe light
lol
fossil kiln
THIS WINNIE
fringe light
lol
worldly mesa
oh you meant start from 0 like this
fringe light
start from 1
worldly mesa
**ch3rry**
I mean here both sums should start from 0
fossil kiln
How does winnie like 1 wot
worldly mesa
How does winnie like 1 wot
its written
fossil kiln
Its prime factors arent 13 and 7?
worldly mesa
Its prime factors arent 13 and 7?
its 7^0 * 13^0
fossil kiln
its 7^0 * 13^0
Tht counts?🤦♂️
fringe light
0 is illogical
worldly mesa
0 is illogical
I am talking about starting from 0 as an index, not as the reciprocal of zero
fickle junco
zero isnt positive
you have to count 0
fickle junco
7^0 is 1
fringe light
one multiplied to and is of no use
fossil kiln
I dont get it
fringe light
yo how do i ask for help
fossil kiln
How does tht count
fickle junco
**ch3rry**
you have:
7^0 (1, 13, 13^2 ...)
7^1 (1, 13, 13^2, ...)
....
which simplifies to this starting from 0 index
fringe light
fossil kiln
No i get tht
But idk feels like cheating
One can also be written as 3^0 so 13,7 arent its only prime factors?
fringe light
The number 1 has no prime factors (or an empty set of prime factors), which is consistent with having only 7 and 13 as factors (vacuously true). In the mathematical formula
fossil kiln
Idt i get it....
Winnie should js get herself some pizza and go to sleep
,w 1/((1-1/7)(1-1/13))
soft zealotBOT
fossil kiln
So this should be the ans??????
fickle junco
One can also be written as 3^0 so 13,7 arent its *only* prime factors?
it does not mean the number must actually contain 7 or 13, it means if it has any prime factors, they must be from the set {7, 13}
fossil kiln
it does not mean the number must actually contain 7 or 13, it means if it has an...
.....what
fickle junco
i think, cause its only if and not if
fossil kiln
I dont get it💔
fickle junco
since 1 has no prime factors, it automatically satisfies “all of its prime factors are 7 or 13"
fickle junco
https://en.wikipedia.org/wiki/Vacuous_truth explains ts
rustic crescent
since 1 has no prime factors, it automatically satisfies “all of its prime facto...
the way you phrased it is how to make it imply that 1 doesnt apply, the interpretation to make 1 valid would be more like "it cannot have prime factors besides 7 or 13", no?
fossil kiln
https://en.wikipedia.org/wiki/Vacuous_truth explains ts
Kay i kindof got it. But what abt only 7 and 13?
Acc Ima js accept it bcz the ques said so💔
fickle junco
the way you phrased it is how to make it imply that 1 doesnt apply, the interpre...
yeah, tbh the question's wording is bad but 1 is an example so im guessing thats what its trying to say
rustic crescent
yeah, tbh the question's wording is bad but 1 is an example so im guessing thats...
what kinda sicko even tries to incorporate 1 into a question about prime numbers anyway lmao
turning this 7th grade math class into a philosophical debate
fossil kiln
what kinda sicko even tries to incorporate 1 into a question about prime numbers...
Ikr winnie needs to go to the psych ward
Should i close now?
.close
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dusk sky
Hi anyone knows about Factor A in parable (parabola or called in english idk) calculate Facor A based in parabola in this topic range
final saddleBOT
Hi anyone knows about Factor A in parable (parabola or called in english idk) ca...
dusk sky
its like point 19 / 3
19 = a x 3(2) + 2 x 3 + 1
19 = 9a + 6 + 1
19 = 9a + 7
9a = 12 a = 12/9 = 4/3
point (3,19)
y=ax2+2x+1
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errant oasis
<@&268886789983436800>
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loud sundial
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fallen valve
I'm a bit confused on Zorn's Lemma + AC and why we given a collection of non-empty sets, why can we not just say pick an element from each non-empty set? Why is this not allowed?
final saddleBOT
I'm a bit confused on Zorn's Lemma + AC and why we given a collection of non-emp...
fallen valve
As in ig it seems it needs to be an axiom i think?
or like Zorn's Lemma/Axiom of Choice is an axiom that cant be proven right
trail mango
sure
fallen valve
I'm a bit confused on Zorn's Lemma + AC and why we given a collection of non-emp...
or sorry not allowed isnt good terminiology, ig i mean why cant we just ppick an elmetn from each non-empty set
it seems there is alot of machienry going on beforehand
proper dagger
"why can't we pick an element from each nonempty set" is exactly what AoC specifies you can do, isn't it?
jagged wave
ac is indepedent from the axioms of zf
if you're working in zf then there are sets for which no choice function exists but aoc resolves this as an axiom
trail mango
or sorry not allowed isnt good terminiology, ig i mean why cant we just ppick an...
your question seems a little more philosophical than mathematical but i can't tell exactly what it is
fallen valve
if you're working in zf then there are sets for which no choice function exists ...
I see okay. I guess my main confusion is how can there be such sets? Since this probably seems dumb, but i cant get over the idea where can just choose an element from each set right?
final saddleBOT
I see okay. I guess my main confusion is how can there be such sets? Since this ...
proper dagger
some families of sets have no rules to them
plucky rover
I see okay. I guess my main confusion is how can there be such sets? Since this ...
Axiom of choice is a weird thing. Let's say you have arbitrarily many sets. Let's say these sets are all uncountably infinite
proper dagger
and in math you need to specify how you're choosing an element out of each set in such families, but because they have no rules, you can't point and choose one
plucky rover
Now it seems like the Cartesian product of these should be non empty intuitively right?
plucky rover
Now it seems like the Cartesian product of these should be non empty intuitively...
Well, give me an element from this Cartesian product
Now it seems easy to say "just pick one element from each set", but that isn't specific enough
You could say "pick the first element" but the sets need not be ordered
You can do maths without AoC to quite a reasonable extent. However you stop having certain things you take for granted
final saddleBOT
@fallen valve Has your question been resolved?
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timid peak
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tired walrus
image 2: the limit of 2/(3n) as n approaches +∞ is not +∞.
little python
yes
final saddleBOT
hasty mist
blud
little python
trail mango
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Available help channel!
forest bobcat
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formal trail
vapid viper
why dox yourself like this
forest bobcat
This is the best i can find from my uni recent paper exams
proper dagger
looking past the test date, you have not stated what you want helpers to do with your past year paper.
forest bobcat
Solve problems
proper dagger
!noans.
forest bobcat
Step by step
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The purpose of this server is to help you learn; please don't ask for direct answers. Ask for guidance, explanations, or feedback instead.
forest bobcat
Can anyone help?
proper dagger
plus, while I definitely am in no position to help with this, for future helpers, show what you understand about the problems, show what you've tried, and tell us what you do not understand.
forest bobcat
Where can i find help with this ?
vapid viper
Where can i find help with this ?
do the problems and come to the server when you are stuck somewhere
forest bobcat
But i dont know how to solve it
storm karma
Translate paper to english first
proper dagger
you don't need to know how to solve it. you need to have tried, or tell us exactly what you do not get about which question
final saddleBOT
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quasi ivy
What is Circumference?
final saddleBOT
What is Circumference?
heady moon
What is Circumference?
cicumference of what ?
quasi ivy
Circle
hidden marlin
What is Circumference?
Its the perimeter of the circle.
quasi ivy
Its the perimeter of the circle.
?
quasi ivy
it is not perimeter i think so
hidden marlin
It is perimeter.
heady moon
it is not perimeter i think so
circumference is a fancy way to say perimeter of circle
hidden marlin
Yeah.
Since a perimeter is just the length of the thing which encloses a finite amount of area.
Circumference is a perimeter.
Because if you look at a circle. The area is enclosed by a outline'. Since a certain, finite area is enclosed by a thing. It must be the perimeter. And for a Circle, this perimeter is also called circumference
quasi ivy
yes
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quasi ivy
What is formula used to find circumference of circle
final saddleBOT
What is formula used to find circumference of circle
worthy wren
2πr
smoky yoke
That was my brother
severe canyon
?
What's your doubt? That IS the answer to your question!
Do not offend!
<@&268886789983436800>
worldly spruce
Family reunion
smoky yoke
👍
quasi ivy
C = 2πr (using radius) or C = πd (using diameter). Both formulas calculate the distance around the circle, where the diameter is twice the radius ((d=2r)).
smoky yoke
✅
worldly spruce
C = 2πr (using radius) or C = πd (using diameter). Both formulas calculate the d...
Yes
And that was your question
"the distance around the circle" $\equiv$ circumference
soft zealotBOT
USS-Enterprise
quasi ivy
yes
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sacred night
can someone explain the solution?
final saddleBOT
can someone explain the solution?
sacred night
MP is the multiplication principle and AP is the addition principle
why is it (100-k)^2?
and not k(100-k)^2
winter halo
and not k(100-k)^2
they fixed a as k, so there are 100-k choices for b and 100-k choices for c, and then they applied addition principle over k=1, 2, ...99
sacred night
they fixed a as k, so there are 100-k choices for b and 100-k choices for c, and...
but why is k not multiplied
with (100-k)^2
because the multiplication principle is
,,\prod_{i=1}^r n_i
soft zealotBOT
calvin
sacred night
for some event ‘E’ decomposed into ‘r’ ordered events
anyoneee
<@&286206848099549185>
sacred night
how can i help u
can you explain the solution please
i dont understand why its not k(100-k)^2
last quartz
where which one
sacred night
where which one
scroll up
last quartz
sorry i havrnt been really active cuz my mom did surgery n im taking care of her