Limits, geometric series, cauchy, proof HELP

[chrisduluk]

but where did i write that P(0)=a+ar+...+ar^n?

 

[Fredrik]

but where did i write that P(0)=a+ar+...+ar^n?

Sorry, you wrote P(n)=, not P(0)=. But it's still of the form "statement=number". It's on line 2 of the handwritten stuff in post #23. Then you did something similar on line 7.

 

[chrisduluk]

i got this proof from class lecture notes. look here on page 13

http://www.cse.yorku.ca/course_archive/2008-09/S/1019/Website_files/15-mathematical-induction.pdf

where is it wrong? This is exactly how I've written all my induction proofs...

 

[Fredrik]

The presentation says e.g. that "P(n) is 1+2+…+n= n(n+1)/2". (This is on page 5). There's nothing wrong with that. It would however be wrong to replace the word "is" with an equality sign, because that would mean that P(n) is the number 1+2+...+n and the number n(n+1)/2, when it's supposed to be the statement 1+2+…+n= n(n+1)/2. It would make sense to write something like
$$P(n)=\big[1+2+\cdots+n= n(n+1)/2\big]$$ but I wouldn't recommend that either. Just say it in plain English: For each positive integer n, P(n) is the statement $1+\cdots+n= n(n+1)/2$.

 

[chrisduluk]

ohhh i see! So does this look a little better?

 

[chrisduluk]

OK, I've started working on the cauchy proof now. My teacher wants us to use the .71717171 to prove that our Sn converges to our "guess".

So this is what we have:

 

[chrisduluk]

i started working on the cauchy proof. how does this look? I don't know how to find big N though, can you help?

 

[chrisduluk]

anybody?

 

[Fredrik]

ohhh i see! So does this look a little better?

Yes, but you still have "P(k+1)=" on line 7.

OK, I've started working on the cauchy proof now. My teacher wants us to use the .71717171 to prove that our Sn converges to our "guess".

So this is what we have:

Looks like you part II (a) under control, and the part of II (b) that isn't a convergence proof.

i started working on the cauchy proof. how does this look? I don't know how to find big N though, can you help?

The Cauchy stuff is problem I (b). Here you just want to find the sum of the series. I would prefer to prove the formula for the sum of a geometric series, and only insert 0.7171... in the final result, but your approach works too. It might be easier as well.

To find the appropriate N, start by solving the equality you have (involving n and ε) for n. Keep in mind that your N can depend on ε (because the definition of "limit" says "For all ε>0, there's an integer N such that...").

Edit: You found an inequality that you want to be satisfied for all n such that n≥N. That means that in particular, it needs to be satisfied when n=N. This is why the inequality tells you something about N. However, my "solve for n" tip may not have been so great. I need a few minutes to think about it.

Also keep in mind that I'm not checking all the details of your calculations.

 

[chrisduluk]

how do i do that? do you mean "in"equality?

 

[Fredrik]

how do i do that? do you mean "in"equality?

Yes, I meant inequality.

 

[chrisduluk]

i know you said you're not checking my calculations, but could you on this one? I'm rusty...

And if it's right... what now?

EDIT: see next page, page 4 for more posts

 

[Fredrik]

I had to think about this, because I had a memory of a trick that I've used to prove something like this before, and I thought it might make things easier. It turned out to be a dead end, so solving for n is indeed the way to go. I think you got the inequality wrong by a factor of 10 (before you started solving for n), so you might want to have another look at that. Also, shouldn't the exponent be n+1? Edit: Hm, maybe those two differences cancel each other out. Let me think...yes they do. Never mind then.

The rest of it looks good, but I suggest that you use the notation $\log_{10} x$ or $\lg x$ for the base-10 logarithm. Note that you still need to choose your N, and it needs to be an integer. (It's OK to do that with a statement of the form "let N be any integer that's greater than X").

 

[chrisduluk]

oh yeah, it should be 9900 not 990

 

[chrisduluk]

so this is what I'm getting now... is this right? How do i use this to find big N?

 

[Fredrik]

I had 99 instead of 990, but I left the exponent as n+1 instead of changing it to n. For a moment I thought those two cancel each other out, but they don't, since $(1/100)^{n+1}=(1/100)^n/100$. Now the two differences between your result and mine cancel each other out.

I don't think there are any more hints I can give you about N. You will have to figure it out on your own. You're almost there. Edit: Hm, some of what I said in #48 was added in an edit. I don't remember if the comment about N is one of those things. You may have missed it.

 

[chrisduluk]

In all of the 5 examples of using cauchy I've ever had, to find N we do...

but I'm not getting a true statement. What do i do?? How do i use what i got for little n to find big N? I've never used that before.

Usually N would look like 1/epsilon, or 245/epsilon, etc... it was never just an integer.

 

[chrisduluk]

ok, take back my last post. How about this??

 

[chrisduluk]

is this ok for my N?

 

[chrisduluk]

and can you help me word this one out?


Part 1(b): Find conditions on a and r such that the sequence of partial sums of $\sum_{k=0}^\infty ar^k$ is a Cauchy sequence if and only if those conditions are satisfied. (Note that this means that the series is convergent if and only if those conditions are satisfied).

 

[Fredrik]

is this ok for my N?

You have to keep in mind that ε may not be an integer (and $\ln 10$ certainly isn't), and N must be an integer. None of the Ns you have suggested are integers.

Don't forget what it is you're trying to do. You're looking for an N such that the following two statements are true:

1. N is a non-negative integer.
2. For all integers n such that n≥N, $\frac{99}{71}(0.01)^{n+1}<\varepsilon$

There is obviously more than one such N.

and can you help me word this one out?Part 1(b): Find conditions on a and r such that the sequence of partial sums of $\sum_{k=0}^\infty ar^k$ is a Cauchy sequence if and only if those conditions are satisfied. (Note that this means that the series is convergent if and only if those conditions are satisfied).

I think this is the hardest part of the problem. I will take a look at it.

 

[chrisduluk]

can you PLEASE help me figure out what my N is supposed to look like? I simply don't know how else to do it other than the way i did it above.

I BEG of you. BEG! I need to finish this!

 

[micromass]

can you PLEASE help me figure out what my N is supposed to look like? I simply don't know how else to do it other than the way i did it above.

I BEG of you. BEG! I need to finish this!

Fredrik is NOT going to give you the answer. You will have to figure it out on your own. We can only guide you to the solution.

 

[chrisduluk]

then help guide me into finding an N that works! Should it have logs in it? Should the N have epsilon in it? Why was my N wrong above?

 

[Fredrik]

I understand that this is important to you, but I don't want to tell you so much that I'm breaking the forum rules. Let me ask you this, is it at least clear to you that an N with the desired properties exist? Can you explain why such an N must exist?

Problem I (b) is easier than I thought it would be, but it's still kind of hard. You have made it harder for me to explain it to you by not answering the questions I asked here:

We need to make sure that you understand a few other things:

1. Do you understand what it means to say that a series is convergent? Specifically, if I write $\sum_{k=0}^\infty a_k=s$, do you know what that means?

2. Do you understand what it means to say that a sequence is convergent? Specifically, if I say that $s_n\to s$, do you know that means?

3. Do you understand the definition of Cauchy sequence? Specifically, if I say that $\langle s_n\rangle_{n=0}^\infty$ is a Cauchy sequence, do you know what that means? (That's the notation I use for the sequence $s_0,s_1,\dots$. Your teacher may use something different).

The work you showed me on problem II (b) suggests that you know the answer to questions 1 and 2. Can you at least answer question 3?

 

[chrisduluk]

i do realize there's an N so it satisfies both of those properties, i just don't know how to eliminate the logs so N is an integer. I've never done an example this ridiculous... All of my examples worked out nicely, so it's clear we'd just make N an integer/epsilon. How do i eliminate the logs when finding my N? Is it simply 71/99E +anything?

and i do understand what makes it cauchy, we pick any positive epsilon, and pick any two values down the line of the sequence and say the difference between the two is whatever we want it to be...

 

[Fredrik]

How do you find the N when you prove that 1/n→0? Let ε>0 be arbitrary. We want to prove that there's a natural number N such that for all integers n, n≥N implies |1/n-0|<ε. If we solve this for n, we get n>1/ε. But 1/ε isn't an integer either.

 

[chrisduluk]

...so what's that mean? huh?

Can you just tell me what part of the N i calculated above is wrong? And how i can make it right? ie remove the denominator, add 1 to it, etc??

 

[Fredrik]

and i do understand what makes it cauchy, we pick any positive epsilon, and pick any two values down the line of the sequence and say the difference between the two is whatever we want it to be...

You're going to have to be much more specific about what a Cauchy sequence is when you start working on I (b). It seems to me that you're supposed to use the following:

Definition: A series is convergent if and only if its sequence of partial sums is convergent. If the sequence is convergent, its limit is called the sum of the series.
Theorem: A series with real terms is convergent if and only if its sequence of partial sums is a Cauchy sequence.

You need to use this theorem to prove that your series is convergent. So the first thing you should write down is exactly what it means for your sequence of partial sums to be a Cauchy sequence.

 

[Fredrik]

How do you find the N when you prove that 1/n→0? Let ε>0 be arbitrary. We want to prove that there's a natural number N such that for all integers n, n≥N implies |1/n-0|<ε. If we solve this for n, we get n>1/ε. But 1/ε isn't an integer either.

...so what's that mean? huh?

You tell me. This is the simplest possible problem of the same sort that you need to solve. So you should probably put your problem aside for a while, and figure out the answer to this one first.

 

[chrisduluk]

Fredrik, I'm sorry but I'm not following you and i don't have any more time to work on this. I HAVE AN EXAM TOMORROW MORNING. I haven't even started studying for it yet because i keep running around aimlessly on this! I specifically posted on here to get HELP and I'm just getting more confused.

Can you PLEASE try to make things more clear to guide me step by step so i can do these problems? Right now i need to know why my N was wrong. Can you follow my work that i scanned above?

 

[chrisduluk]

why can't i make my N= [ln(71/99E) / ln(100)] +1

 

[Fredrik]

Fredrik, I'm sorry but I'm not following you and i don't have any more time to work on this. I HAVE AN EXAM TOMORROW MORNING. I haven't even started studying for it yet because i keep running around aimlessly on this! I specifically posted on here to get HELP and I'm just getting more confused.

I'm not going to break any forum rules because you have an exam tomorrow. I have given you much more information than you would have needed if you had done a few more exercises before you came here. And I have spent a lot of time giving you that information. So don't act like you haven't gotten any help. We have made significant progress, but we would have made more if you hadn't been so unwilling to write down definitions and answer questions.

Can you PLEASE try to make things more clear to guide me step by step so i can do these problems?

I have done that for all parts except what you need to finish I(b), but since you haven't even followed my instructions on how to start I(b) yet, you have no need to for the rest of it yet.

Right now i need to know why my N was wrong. Can you follow my work that i scanned above?

What N are you talking about? The one where you say that N is equal to an inequality? That doesn't even make sense.

 

[Fredrik]

why can't i make my N= [ln(71/99E) / ln(100)] +1

Do the brackets mean something different than parentheses here?

 

[chrisduluk]

on the scanned sheet, ignore the inequality, so N= everything to the right of the inequality sign.

and you mean the floor function?