Prove that $ \frac{1}{15} + \frac{1}{27} + \frac{1}{3*9}+... = \frac{8}{9} - \frac{2}{3}\ln(2) $

Asked Jun 05 '19 at 07:27

Active Jun 05 '19 at 07:53

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Prove that $ \frac{1}{1*5} + \frac{1}{2*7} + \frac{1}{3*9}+... = \frac{8}{9} - \frac{2}{3}\ln(2)$

I got the answer using the hint given in the question It mentioned to consider the integral$\int x^2 \ln(1-x^2)dx$. I evaluated the integral by integration by parts.

Then I again evaluated by applying Taylor series. I equated both expressions and took limits of integration from 0 to 1 to obtain the result.

Is there any other way to solve this question? Without the hint, I would never have thought of considering that Integration.

What I tried: $\sum_{i=1}^{\infty} \frac{1}{i(2i+3)} = \frac{1}{3} \sum_{i=1}^{\infty} \frac{1}{i}-\frac{2}{2i+3}$

But both of the above series diverge

edited Jun 05 '19 at 07:53

asked Jun 05 '19 at 07:27

user600016

2,165

Oh sorry . I have corrected it now – user600016 Jun 05 '19 at 07:53
How do you all identify the question has been posted before?I mean, how to search and find the question? – user600016 Jun 05 '19 at 07:56
1

Have a look at How do you search for duplicates?. For your series, Approach0 quickly finds an identical question. – Martin R Jun 05 '19 at 08:04

Prove that $ \frac{1}{1*5} + \frac{1}{2*7} + \frac{1}{3*9}+... = \frac{8}{9} - \frac{2}{3}\ln(2) $

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Prove that $ \frac{1}{15} + \frac{1}{27} + \frac{1}{3*9}+... = \frac{8}{9} - \frac{2}{3}\ln(2) $