A series that came up while solving an integral

While solving problem 9 of Brilliant Integration Contest Season 1, I did the following -

\[\begin{align}I= \int_{0}^{1} \log \left(\dfrac{1+x}{1-x}\right) \dfrac{1}{x\sqrt{1-x^2}} \ \mathrm{d}x &= 2 \int_{0}^{1}\sum_{n=0}^{\infty} \dfrac{x^{2n+1}}{2n+1} \dfrac{1}{x\sqrt{1-x^2}} \ \mathrm{d}x\\ &=2\sum_{n=0}^{\infty} \int_{0}^{1} \dfrac{x^{2n}}{(2n+1)\sqrt{1-x^2}} \ \mathrm{d}x \end{align}\] Then I used the substitution $x \mapsto \sin \theta $.

$\begin{aligned} \therefore I &=2\sum_{n=0}^{\infty} \int_{0}^{\pi/2} \dfrac{\sin^{2n} {\theta}}{2n+1} \ \mathrm{d}\theta\\ &=\pi \sum_{n=0}^{\infty} \dfrac{(2n)!}{2^{2n}(n!)^2(2n+1)} \end{aligned}$ The last step is due to Wallis' formula.

Now, my question is that how do we evaluate this last series?

#Calculus #Series #InfiniteSum

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Have to changed the limits to obtain the consequences in using wallis formula.

Mark Lawrence Ureta - 6 years, 6 months ago

I didn't get what you are saying.. could you explain?

Pratik Shastri - 6 years, 6 months ago

You are in class 12 and you know such a high level calculus , can you give any suggestions or a recommened a book for increasing my ability in calculas@Pratik Shastri

sandeep Rathod - 6 years, 6 months ago

@Sandeep Rathod – Well, I generally just browse the internet for interesting topics, and Wikipedia pages certainly help.

Pratik Shastri - 6 years, 6 months ago

Well, @Krishna Sharma just posted a similar question as a problem. here. It is not exactly the same but quite a bit yeah. And it has a solution too.

Kartik Sharma - 6 years, 3 months ago

I posted this long ago :P I now know how to get the result.

Pratik Shastri - 6 years, 3 months ago

Can you post the result here ?

Uchiha Robert - 6 years, 3 months ago

@Uchiha Robert – Well, you can use Beta function!

Kartik Sharma - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$