Limits, geometric series, cauchy, proof HELP

[chrisduluk]

i guys, I'm stuck on wording of a homework assignment and thought you might be able to help me. There are several questions...

Consider the geometric series: (Sum from k=0 to infinity) of ar^k
and consider the repeating decimal .717171717171 for these problems:

Question 1:
Find a formula for the n-th partial sum of the series and PROVE your result using the Cauchy Convergence Criterion. This technique requires to find epsilon, n, N, etc…

Question 2:
Use your formula from Q1 above to determine which conditions on "a" and/or "r" guarantee that the geometric series converges. And PROVE your result.

Question 3:
Write the repeating decimal .7171717171717171... as a geometric series.

Question 4:
Find the sum of the geometric series in Q3 above to get a fractional representation of your repeating decimal.


My attempts at solving these:

My issues with Q1 are:
- is the formula for the nth partial sum a/(1-r) or [a-ar^(n+1)] / (1-r)?
- the problem notes sum from 0 to infinity; does this change the soln?
- our professor is very picky when it comes to proofs. we have to use the Cauchy criterion to prove this. We can't just use calculus methods to prove things converge such as limit test, ratio test, integral test etc...

My issues with Q2 are:
-How do I prove this??

My issues with Q3 and Q4 are:

I wrote in the margin on my homework page that I'd have to "prove it has equivalent rational representation..." but I don't know what this applies to?

so far I have that .717171… = (sum from k=0 to infinity) of .71(.01)^k
but I'm lost as far as doing the proofs…


Thanks for any help guys! I have today and tomorrow to do this. Thanks for any guidance!

 

[micromass]

Can you state the Cauchy convergence criterion??

 

[chrisduluk]

All the cauchy proofs I have been given begin with:

Let epsilon >0 be given. We want to find N element of the naturals such that n>=N implies that abs[...- thing converges to] < elsilon. Then we need to choose an N and yada yada...

I'm going to be honest when I say I don't get it either...

 

[chrisduluk]

can anybody help me with this?

 

[Fredrik]

I assume that the "Cauchy convergence criterion" is simply that when we want to prove that a sequence of real numbers is convergent, it's sufficient to prove that it's a Cauchy sequence (because we know that every Cauchy sequence is convergent).

You haven't stated the definition of a Cauchy sequence correctly, so I suggest you start with that.

The nth partial sum of a series $\sum_{k=0}^\infty x_k$ is $\sum_{k=0}^n x_k$. If you understand that, you shouldn't have a hard time figuring out which one of your formulas for the nth partial sum of $\sum_{k=0}^\infty ar^k$ is the right one.

The main reason why you haven't received a lot of help is that you didn't do the most obvious thing: Write down the relevant definitions and make an attempt to apply them. The more work you put in, the more help you will get. If you don't do the obvious first step, you are much less likely to get a reply.

 

[chrisduluk]

OK, here's what I've got for the definition of the cauchy criterion. But i don't know how to apply it to these problems. We've done literally 5 examples using cauchy, and i don't understand any of them...

Someone else gave me this but it's all greek to me

 

[Fredrik]

I have a question about your Q1. It says "prove your result". What result? The only thing you're supposed to before that is to find a formula for the nth partial sum, so that formula is the only result you have at that point. And if you have found it, I assume that you have already proved it. It's not the sort of thing you would just guess. I think you need to post the exact statement of the problem.

 

[chrisduluk]

believe me, i agree with you that the questions don't make sense. Here is the exact form we were given. Each student was assigned a different value, mine was .71717171...

And i know we need to use the cauchy criterion to prove something converges to a certain value. Can't just say r < 1 implies it converges.

 

[micromass]

That's something completely different from your OP!

You need to
1) find a formula for the partial sums. (Cauchy criterion has nothing to do with this)
2) prove that the partial sums converge using the Cauchy criterion.

So I suggest you start by finding and proving a formula for the partial sums:

$$\sum_{k=0}^n{ar^k}=...$$

 

[chrisduluk]

Ok, does this help?

or this?

 

[Fredrik]

OK, I think what you're supposed to do in Q1 (book problem I(a)) is to prove that the formula for the nth partial sum that you already seem to be familiar with holds, nothing more. So Q1 has nothing to do with convergence. It's a finite sum. Do you know how to do this? (Please post your solution, not a picture of someone else's).

 

[chrisduluk]

OK, I think what you're supposed to do in Q1 is to prove that the formula for the nth partial sum that you already seem to be familiar with holds, nothing more. So Q1 has nothing to do with convergence. It's a finite sum. Do you know how to do this?

nope. :uhh:

All i know is that the cauchy criterion needs to be used somewhere in this assignment...
And i have to apply the .717171717 to all parts of this assignment, not just the last parts.

 

[chrisduluk]

Yepp, totally lost. Do i do it like these examples?

 

[Fredrik]

nope. :uhh:

All i know is that the cauchy criterion needs to be used somewhere in this assignment...
And i have to apply the .717171717 to all parts of this assignment, not just the last parts.

You're supposed to solve problems I (a)-(b) and II (a)-(b), right? I see no reason to think that .7171... should be involved in problem I, and no reason to think that the Cauchy criterion should be involved in problem I (a). Problem I (a) isn't even about a series. It's a finite sum.

Problem I (a) is about one thing: You need to rewrite $1+x+x^2+\cdots+x^n$ in a simpler form. I will tell you the trick. For all real x such that x≠1,
$$1+x+\cdots +x^n=\frac{(1+x+\cdots+x^n)(1-x)}{1-x}=\cdots$$ Can you take it from here? (The problem you were given was very slightly different, but you should be able to figure out how to deal with that difference).

Yepp, totally lost. Do i do it like these examples?

Your solution of II (a) should look like those examples.

 

[chrisduluk]

I guess you're right about not using the repeating decimal .717171 and cauchy for part 1. I have no idea...

I hate to sound like an idiot but i can't follow where you're trying to go... :/

EDIT: I posted again, we went to a second page now...

 

[Fredrik]

Please multiply those two factors in the numerator together.

Did it not occur to you to try that? You don't always have to see where a calculation is going when you start. Sometimes you just have to try something and see where it takes you.

 

[chrisduluk]

so [1-x^(n+1)] / (1-x) ?

 

[chrisduluk]

hmm... why am i still lost?

I really don't mean to sound like a dolt, i just don't understand what the answers are even supposed to "look" like. My teacher is beyond terrible and can't explain anything...

and when do i use cauchy??

 

[Fredrik]

so [1-x^(n+1)] / (1-x) ?

That's the final result yes. But how did you get that? When you just write down the final result, I have no idea if you actually did the multiplication, and neither will your teacher when you hand it in. (Are you supposed to hand it in?)

hmm... why am i still lost?

I really don't mean to sound like a dolt, i just don't understand what the answers are even supposed to "look" like. My teacher is beyond terrible and can't explain anything...

In this case, I think problem II (b) explains the reason why you're doing these things. The goal is to prove that a repeating decimal expansion can always be expressed as a fraction n/m, where n and m are integers. The other problems can be thought of as intermediate results on the way there, that are also supposed to give you some practice with geometric series and Cauchy sequences, both of which are very important concepts on their own.

I think the reason why you feel completely lost is that you are so unwilling to try anything unless you know that the result will be what you want. This seems to really be holding you back. Sometimes you just have to ask yourself "is there any way I can rewrite this expression", and then just do it to see if the result is easier to work with.

and when do i use cauchy??

In problem I (b). That looks like the hardest part. I think someone else will have to help you with that, because I'm going to bed soon. But even if no one else shows up for a while, don't let that stop you from trying out some ideas.

By the way, you can't be sure that your teacher won't read your posts here.

 

[chrisduluk]

I contacted my teacher, and this is what they said:

You need to prove that your partial sum is correct (I cannot comment whether it
is or not) -- that is, you will need to use induction. And then you need to
prove that the limit of your partial sum converges to your guess. That is, you
will need an appropriate epsilon-N proof. and yes indeed, we need to write up these answers and hand them in. This is actually a project that's a big portion of our grade. After this i'll never see math again! Just need to get these problems done so i can move on...!

So my questions now are, what is the partial sum in 1a? I could probably do the induction to prove it, if i was only clear as to what the partial sum was... And for 1b, how do i use cauchy? I'm being honest when i say i don't know how to even begin.

 

[Fredrik]

The multiplication we did for I (a) can be considered non-rigorous, because even though it looks obvious, we didn't really prove that all those other terms cancel each other out*. That's why you have to use induction to prove the formula we found, or rather to prove the formula you will find when you repeat the above calculation with $ar^n$ instead of $x^n$.

*) A mathematician doesn't actually consider that step non-rigorous, because he immediately knows that he could prove it by induction if he wanted to. Students with less experience should however do every step for pedagogical reasons, so I think your teacher is right to demand it here.

 

[Fredrik]

So my questions now are, what is the partial sum in 1a? I could probably do the induction to prove it, if i was only clear as to what the partial sum was...

You don't need to be 100% sure that you have the right formula, because if you have the wrong one, the induction proof can't possibly succeed, and the failure should make you reconsider the formula.

And for 1b, how do i use cauchy? I'm being honest when i say i don't know how to even begin.

The very first thing you need to do is to study the definition of a Cauchy sequence, and make sure you understand it.

 

[chrisduluk]

OK, for the induction, can you at least set it up for me? If the questions are "worded right" i'll know what to do...
For example, if you ask me to prove that 1 + x + x^2 + ... + x^n = [x^(n+1)-1] / (x-1) using induction, i could do it. So what's the "left side" and "right side" of what i need to do for this assignment?

And as far as using cauchy, I've only done a few problems, and they've all been worded like:
"for the sequence Sn= 1 + 1/(2^n) find the limit and prove your sequence converges to your limit..." So i have NO idea how this assignment is asking me to use cauchy. Where am i taking limits? What's converging?

Do you see what I'm trying to say?

So is this right for the induction part?

And i know that my Sn converges to a/(1-r) ...is this were I'm supposed to use cauchy to prove it? If so, i don't know how to do so. Can you set it up for me?

 

[Fredrik]

The induction part of it is essentially correct. It can be better stated though. Remember that the point of an induction proof is that it allows you to prove infinitely many statements in a finite number of steps. 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.

I have to ask, did you actually do the multiplication I asked you to do? $$(1+x+\cdots+x^n)(1-x)=?$$ Even if you did, I suggest that you do it again.

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).

 

[Fredrik]

By the way, if you have to post pictures, try to keep them smaller. It's better if you type the mathematics. Click the quote button next to one of my posts, and you'll see how i did the math there. There's a LaTeX guide for this forum. You just need to read a small part of it to get started.

This is how I would state the problems you're supposed to do:

I
(a) Find a formula for the nth partial sum of $\sum_{k=0}^\infty ar^k$, and use induction to prove that it holds for all non-negative integers n.
(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).
(c) Use the formula from I(a) to guess a simple formula for $\sum_{k=0}^\infty ar^k$ that holds when the conditions on a and r from II(b) are satisfied. Prove that the sum of the series is in fact what you guessed.

(Yes, I know that there's no I(c) in your book, but it certainly looks like this is a part of what you're supposed to do, and I can't tell if your teacher considers it a part of I(b) or II(b), so I stated it as a separate problem).

II
(a) Write 0.7171... as a geometric series.
(b) Prove that there exist integers p and q such that 0.7171...=p/q, by using the formula for the sum of a geometric series found in I (c) on the series found in II (a).

Keep in mind that I have to be careful to not tell you too much. I can only give you hints about how to proceed, so the best way to keep this moving along is to always post your work up to the point where you are stuck. If the next step after that is just to apply a definition, you should only expect replies like "use the definition".

 

[chrisduluk]

can you write out what's wrong with my induction proof then? I'm not following what you're saying is wrong with it... I can't find anything where p(0)= what you said was wrong...

I REALLY need you to help me accelerate with these problems. I don't have anymore time to mess around i have an exam to study for!

 

[Fredrik]

I will tell you one thing: The right-hand side of the equality you're trying to prove by induction is wrong. Now, why are you unable to discover that yourself? It must mean that you chose not to follow my suggestion to do the multiplication again. I didn't suggest that you do it again just to mess with you.

This isn't going to work if you are unwilling to do things on your own. You can't just expect me or someone else here to show you the complete solution. Even if we wanted to do that, it would be against the forum rules.

 

[chrisduluk]

you mean the [ar^(n+1)-a] /(r-1) is wrong? isn't that what we came up with earlier?

so that's not the nth partial sum??when i did the multiplication you asked i got [1-x^(n+1)] / (1-x) but i don't know how this relates to the sum of ar^k

 

[Fredrik]

Sorry, I didn't see that you changed the order of the terms in the denominator, so it looked like you had the wrong sign. My mistake.

 

[Fredrik]

Edit: Ignore this post. I wrote this before I realized that I had made a mistake. (See the post above this one).

you mean the [ar^(n+1)-a] /(r-1) is wrong? isn't that what we came up with earlier?

so that's not the nth partial sum??


when i did the multiplication you asked i got [1-x^(n+1)] / (1-x) but i don't know how this relates to the sum of ar^k

You're supposed to find a formula for $\sum_{k=0}^n ar^k$ that holds for some values of r, and you should be able to figure out which values of r that is when you find the formula. I told you how to find a formula for $\sum_{k=0}^n x^k$ that holds for all x such that x≠1. Do these two formulas look like they have nothing to do with each other?

You have two options:

1. Use the formula for $\sum_{k=0}^n x^k$ to find to find the formula for $\sum_{k=0}^n ar^k$.
2. Use the trick we used to find the formula for $\sum_{k=0}^n x^k$ to find a formula for $\sum_{k=0}^n ar^k$.

 

[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.