Limits, geometric series, cauchy, proof HELP

[chrisduluk]

i don't think i have to show my work as to "how" i got the partial sums formula. I just have to use induction to prove my guess "is" the partial sums formula.

I do see that all we do is multiply the fomula for x^k by a though.,.,

Is the partial sums formula what i have pictured below? It can be written in many ways... (i don't have time to type everything into this forum using the commands, etc. I'm REALLY rushed today...

 

[micromass]

i don't think i have to show my work as to "how" i got the partial sums formula. I just have to use induction to prove my guess "is" the partial sums formula.

I do see that all we do is multiply the fomula for x^k by a though.,.,

Is the partial sums formula this? It can be written in many ways... (i don't have time to type everything into this forum using the commands, etc. I'm REALLY rushed today...

If you would follow Fredriks hint, then you would know whether this was the correct formula or not.

 

[Fredrik]

Yes, that's the correct formula. (I feel that I have to say that, after incorrectly suggesting that it's not). I saw that you had changed the order of the terms in the numerator, but for some reason I didn't see that you had also changed them in the denominator, so it looked like you had the wrong sign. The induction proof is fine too, if you just avoid notational weirdness like "statement=number".

 

[chrisduluk]

what do you mean by statement= a number?

 

[Fredrik]

what do you mean by statement= a number?

I meant what I said here:

Each P(n) is a statement, and you want to prove that they're all true. Induction allows you to do that by proving only two statements:

1. P(0) is true.
2. For all non-negative integers n, if P(n) is true, then so is P(n+1).

You have the right idea about how to do this sort of thing, but the notation is sometimes ugly. For example, you wrote P(0)=a+ar+...+arⁿ. That doesn't make sense, because it says that a statement is equal to a number. Kind of like saying that an elephant is equal to a Tuesday.

 

[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?