Hi,

I am trying to prone that the sum of the following series is log3.

1+1/2-2/3+1/4+1/5-2/6+1/7+1/8-2/9+1/10+1/11-2/12...

Thank's in advance.

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- December 30th 2013, 11:29 AMhedisum of alternating series
Hi,

I am trying to prone that the sum of the following series is log3.

1+1/2-2/3+1/4+1/5-2/6+1/7+1/8-2/9+1/10+1/11-2/12...

Thank's in advance. - December 30th 2013, 02:59 PMchiroRe: sum of alternating series
Hey hedi.

Hint: Look at the taylor series expanded around a = 1 (i.e. taylor series for log(1+x)). - December 30th 2013, 03:06 PMhediRe: sum of alternating series
This gives the harmonic alternating series,it is not so helpfull here.

- December 30th 2013, 03:21 PMchiroRe: sum of alternating series
Can you collect terms together to show that both are the same series?

- December 30th 2013, 03:25 PMromsekRe: sum of alternating series
- December 31st 2013, 05:43 AMSlipEternalRe: sum of alternating series
Let be the -th Harmonic number (the sum of the first terms of the Harmonic series). Consider partial sums of three terms of your series at a time. Show that gives the correct partial sums for your series. So, your series is given by .

Next, use common inequalities for the Harmonic numbers. For example, Young (1991) showed that . So,

and

Adding these inequalities together gives:

Simplifying gives:

Finally, use the Squeeze Theorem to show that , so - December 31st 2013, 07:31 AMSlipEternalRe: sum of alternating series
Btw, to show that , you can write your series as follows:

- December 31st 2013, 08:03 AMhediRe: sum of alternating series
Thanks to both of you so much.

- December 31st 2013, 11:50 AMromsekRe: sum of alternating series
I'm curious what course (if any) this would be learned in. I've had a fair bit of math and never ran into a detailed analysis of the Harmonic series like this.

Is this something you just naturally come across post-grad in the course of solving other problems? - December 31st 2013, 01:06 PMSlipEternalRe: sum of alternating series
- December 31st 2013, 01:13 PMhediRe: sum of alternating series
This question comes from a course in problem solving.

- December 31st 2013, 01:46 PMSlipEternalRe: sum of alternating series
Actually, I just thought of a more direct way to show that the series converges to . By definition, , so

- January 1st 2014, 04:14 PMProve ItRe: sum of alternating series
This is not an alternating series...