- #1

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I'm stuck on this problem, I can't solve it

Sum(from 1 to infinity) of 8 / x^2(4+ln(x))

Does anybody have the fainest idea of how to solve?

Thanks all

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- Thread starter rick906
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- #1

- 12

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I'm stuck on this problem, I can't solve it

Sum(from 1 to infinity) of 8 / x^2(4+ln(x))

Does anybody have the fainest idea of how to solve?

Thanks all

- #2

CRGreathouse

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- #3

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- #4

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- #5

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Oh sorry,it is a sum, not an integral, I don't know why I put x instead of n.

I'm not sure to understand what you people mean by 'integrate and then apply the sum'

Technics I have learned for solving a sum are : Reinman (p-series), the integral test, comparision test, ratio test, the n^th root test and those for alternating series.

I've tried with all of them... nothing worked.

However, the one that seemed the most possible was the integral test (was that what you were talking about?). So, I've checked that f(x)>0, f(x) is continuous and that it decreases at a point. Unfortunately, I'm not able to do the integral... even mathematica gives me strange things (see attachment).

Anyone's got a lead? Or maybe a technic I'm not aware of?

Thanks

I'm not sure to understand what you people mean by 'integrate and then apply the sum'

Technics I have learned for solving a sum are : Reinman (p-series), the integral test, comparision test, ratio test, the n^th root test and those for alternating series.

I've tried with all of them... nothing worked.

However, the one that seemed the most possible was the integral test (was that what you were talking about?). So, I've checked that f(x)>0, f(x) is continuous and that it decreases at a point. Unfortunately, I'm not able to do the integral... even mathematica gives me strange things (see attachment).

Anyone's got a lead? Or maybe a technic I'm not aware of?

Thanks

- #6

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Comparison test works here.

- #7

CRGreathouse

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- #8

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You said that you wanted to solve the sum and not an integral (but then contradicted yourself by posting an integral from mathematica!!!!) but then went on to mention convergence tests???

Look did you really mean to say that you wanted to decide if the series converges or not?

- #9

CRGreathouse

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Here's the bound I have (check this before using!) for f(x) strictly decreasing:

[tex]\int_n^{\infty}f(x)dx<\sum_{k=n}^{\infty}f(k)<f(n)+\int_n^{\infty}f(x)dx[/tex]

So take the sum up to n-1 for some n, then add the integral; that gives the lower bound, and adding f(n) gives the upper bound.

- #10

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Ouh, I was so nervous that I forgot to mention : yes, I'd like to know if the sum converges or diverges.Look did you really mean to say that you wanted to decide if the series converges or not?

As for CRGreathouse, I haven't learned that technic, so I'm not sure if the teacher will allow me to use it in a test.

Vid says

I've tried it many times, can you please tell me what have you compared it to.Comparison test works here.

Thanks again all, I appreciate your fast replies.

Rick

- #11

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8/(x^2(4+ln(x)) < 8/x^2

- #12

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Ouhhh

It's so true. I had totally forgotten that p-series with p>1 converges (I thought it was the opposite!!)

So I think the problem is solve thanks to you all (especially Vid) and I've scanned the problem, if anyone could just check if everything's alright, it'd be really great (attachment).

Thanks (infinite thanks, that is)

Rick

It's so true. I had totally forgotten that p-series with p>1 converges (I thought it was the opposite!!)

So I think the problem is solve thanks to you all (especially Vid) and I've scanned the problem, if anyone could just check if everything's alright, it'd be really great (attachment).

Thanks (infinite thanks, that is)

Rick

- #13

CRGreathouse

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- #14

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I wonder how you knew it from the beginning...maybe something I have not learned yet...

By the way, I really do not understand those signs... maybe this is the notion I do not know.

Anyway, are my steps and my result right?

Thanks again dudes.

- #15

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http://en.wikipedia.org/wiki/Riemann_zeta_function

- #16

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Thanks, it confirms that this is something we haven`t learned (not yet, anyway).

I find something strange with the Riemann zeta function: it says that:

[itex]

\zeta(1)

[/itex] = infinity ok we know this one because it's the harmonic serie (p`s equal or smaller than 1 diverge)

but then, [itex]

\zeta(0.5) = 1.46...

[/itex] is that not supposed to be a p-serie with p=0.5 and, therefore, diverge?

This really confusses me.

Otherwise, if you could confirm my steps (about 2 posts of mine ago), that would really make me feel more confident about that.

Thanks again to all of you.

I find something strange with the Riemann zeta function: it says that:

[itex]

\zeta(1)

[/itex] = infinity ok we know this one because it's the harmonic serie (p`s equal or smaller than 1 diverge)

but then, [itex]

\zeta(0.5) = 1.46...

[/itex] is that not supposed to be a p-serie with p=0.5 and, therefore, diverge?

This really confusses me.

Otherwise, if you could confirm my steps (about 2 posts of mine ago), that would really make me feel more confident about that.

Thanks again to all of you.

Last edited:

- #17

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[tex]\zeta(2 + \epsilon) = \sum_{n=1}^{\infty}\frac{1}{n^{2+\epsilon}}[/tex]

Now expand the summand to first order in epsilon.

- #18

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- #19

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Count Iblis, I would have done what you are saying if I had understood it :rofl: In fact, I don`t know that zeta series quite well now (learned about it 5 mins ago ) But hey, thanks anyway !

Also, have you people check my steps (about 3 posts back now), it's an attached image.

Thanks Thanks Thanks.

- #20

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Thanks

- #21

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Thanks

I thought your solution was excellent. Correct reasoning, step by step logic, no holes, presented linearly from the beginning. It's fantastic. The only thing was that you could have written "by the comparison test..." but I think it's good.

- #22

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Just one thing: you said you 'Thought' it was correct... did you mean you 'think' it's correct or that you thought it was but then you noticed a mistake?

Lol, I know it was linearly presented, but usually it just ain't that way... I made an effort of presentation on this one :rofl:

Thanks again

- #23

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No mistake it's fine.

- #24

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I'd like to thank everyone for helping me solving this problem (which doesn't look hard anymore).

Hope that I'll be able to help other people on this forum.

- #25

CRGreathouse

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[tex]\zeta(2 + \epsilon) = \sum_{n=1}^{\infty}\frac{1}{n^{2+\epsilon}}[/tex]

Now expand the summand to first order in epsilon.

I'm curious. How do you get from the crude [tex]n^\epsilon[/tex] to [tex]\log n[/tex]?

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