\[ \large { \pi }^{ 2 }=\frac { 128 }{ 9 } -128\displaystyle\sum _{ n=1 }^{ \infty }{ \frac { n+1 }{ (2n+1)(2n-1)(2n+3)^{ 2 } } } \]

I was playing with the expansion of \(t^2\sqrt{1-t^2}\) and I came to the series above, can you prove it?

\[ \large { \pi }^{ 2 }=\frac { 128 }{ 9 } -128\displaystyle\sum _{ n=1 }^{ \infty }{ \frac { n+1 }{ (2n+1)(2n-1)(2n+3)^{ 2 } } } \]

I was playing with the expansion of \(t^2\sqrt{1-t^2}\) and I came to the series above, can you prove it?

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TopNewestHere is a quick method.

Using partial fractions, we have,

\(\displaystyle \sum_{n=0}^{\infty} \dfrac{n+1}{(2n+1)(2n-1)(2n+3)} = \sum_{n=0}^{\infty} \left[-\dfrac{1}{16(2 n+1)} + \dfrac{1}{64(2 n+3)} - \dfrac{1}{16 (2 n+3)^2} + \dfrac{3}{64 (2 n-1)} \right] \)

By Regularization, we have,

\(\displaystyle = \dfrac{1}{32} \psi\left(\dfrac{1}{2}\right) - \dfrac{1}{128} \psi\left(\dfrac{3}{2}\right) -\dfrac{1}{64} \psi^{(1)} \left(\dfrac{3}{2}\right) + \dfrac{3}{2} \psi \left(-\dfrac{1}{2}\right) \)

Note that,

\(\displaystyle \psi \left(\dfrac{1}{2}\right) = -\gamma - \ln 4\)

\(\displaystyle \psi \left( \dfrac{3}{2} \right) = 2 - \gamma - \ln 4\)

\(\displaystyle \psi^{(1)} \left(\dfrac{3}{2}\right) = \dfrac{\pi ^2}{2} - 4 \)

\(\displaystyle \psi \left(-\dfrac{1}{2} \right) = 2 - \gamma - \ln 4\)

\(\displaystyle \implies \sum_{n=0}^{\infty} \dfrac{n+1}{(2n+1)(2n-1)(2n+3)} = - \dfrac{\pi^2}{128}\)

\(\displaystyle \implies \sum_{n=1}^{\infty} \dfrac{n+1}{(2n+1)(2n-1)(2n+3)} = \dfrac{1}{9} - \dfrac{\pi^2}{128} \) – Ishan Singh · 1 year, 4 months ago

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– Ariel Gershon · 1 year, 4 months ago

Could someone please explain what \(\gamma\) is? I've seen it used in a lot of places without a definition. Is it a variable to represent infinity? Because the series \(\displaystyle\sum_{n=0}^{\infty} \frac{-1}{16(2n+1)}\) definitely diverges, for instanceLog in to reply

@Ariel Gershon Note that the individual sums are not equal to the digamma values I wrote. They are assigned that value, i.e, if the individual diverging sums appear along with some other diverging series such that their value (taken together) is converging, then we can evaluate them by their assigned values. It is similar to assigning a value of \(-\dfrac{1}{12}\) to \(\zeta(-1)\), which in fact, diverges. \(\gamma\) is the Euler - Mascheroni constant which is equal to \(- \psi (1)\). – Ishan Singh · 1 year, 4 months ago

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– Aditya Kumar · 1 year, 4 months ago

It is euler-mascheroni constant.Log in to reply

– Ariel Gershon · 1 year, 4 months ago

Ok thanksLog in to reply

– Aditya Kumar · 1 year, 4 months ago

Yeah even I did it the same way. I had posted a similar problem which uses digamma.Log in to reply

– Hummus A · 1 year, 4 months ago

nice proof!Log in to reply

@Hummus a How can we solve it using series expansion? (As you have stated) – Samuel Jones · 1 year, 4 months ago

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– Hummus A · 1 year, 4 months ago

and i now remember i used \({ x }^{ 2 }\sqrt { 1-{ x }^{ 2 } } ={ x }^{ 2 }-\displaystyle\sum _{ n=2 }^{ \infty }{ \frac { \begin{pmatrix} 2n-4 \\ n-2 \end{pmatrix} }{ (n-1){ 2 }^{ 2n-3 } } { x }^{ 2n } } \)Log in to reply

– Hummus A · 1 year, 4 months ago

you can integrate its expansion,then let \(x=\sin{t}\) then integrate from 0 to pi/2(i think,it's some definite integral,and i'm not very organized with my work,i tend to do it at any piece of paper i have in front of me, yes i am very lazy),then do a lot of tedious arranging,then get it,it's the same technique i used in my second expansion of \(\pi^2\)Log in to reply

Here is a way to solve it without digamma functions:

We start out with the partial fraction expansion: \[128\sum_{n=1}^{\infty} \frac{n+1}{(2n+1)(2n-1)(2n+3)^2} = \sum_{n=1}^{\infty} \left(-\frac{8}{2n+1} + \frac{6}{2n-1} + \frac{2}{2n+3} - \frac{8}{(2n+3)^2}\right)\]

We can split this into three different sums: \[= \sum_{n=1}^{\infty} \left(\frac{8}{2n-1}-\frac{8}{2n+1}\right) - \sum_{n=1}^{\infty}\left(\frac{2}{2n-1} - \frac{2}{2n+3}\right) - \sum_{n=1}^{\infty}\frac{8}{(2n+3)^2}\]

The first sum is a telescoping sum: \[ \left(\frac{8}{1}- \color{red}{\frac{8}{3}}\right) + \left(\color{red}{\frac{8}{3}}-\color{green}{\frac{8}{5}}\right) + \left(\color{green}{\frac{8}{5}}-\frac{8}{7}\right) + ...\] Every term gets cancelled except the first one. Therefore the first sum is \(8\).

The second one is also a telescoping sum, but it's a little more tricky. The cancellation starts happening at the third term, so the result is the sum of the positive parts of the first two terms: \[\left(\frac{2}{1} - \color{red}{\frac{2}{5}}\right) + \left(\frac{2}{3} - \color{orange}{\frac{2}{7}}\right) + \left(\color{red}{\frac{2}{5}} - \color{green}{\frac{2}{9}}\right) + \left(\color{orange}{\frac{2}{7}} - \frac{2}{11}\right) + \left(\color{green}{\frac{2}{9}} - \frac{2}{13}\right) + ...\] Only the terms \(\frac{2}{1}+\frac{2}{3}\) don't get cancelled out. Therefore the value of the second sum is \(\frac{8}{3}\).

Now let's find the last sum, using the fact that \(\sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}\): \[\sum_{n=1}^{\infty}\frac{8}{(2n+3)^2} = 8\left(\frac{1}{5^2} + \frac{1}{7^2} + \frac{1}{9^2} + \frac{1}{11^2} + ...\right)\]\[= 8\left[\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ...\right) - \left(\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \frac{1}{8^2} + ...\right) - \frac{1}{1^2} - \frac{1}{3^2}\right]\]\[ = 8\left(\sum_{k=1}^{\infty} \frac{1}{k^2} - \frac{1}{2^2} \sum_{k=1}^{\infty}\frac{1}{k^2} - 1 - \frac{1}{9}\right)\]\[= 8\left(\frac{\pi^2}{6} - \frac{\pi^2}{4*6} - \frac{10}{9}\right) = \pi^2 - \frac{80}{9}\]

Therefore, the total is: \[\frac{128}{9} - \left[\sum_{n=1}^{\infty}\left(\frac{8}{2n-1}-\frac{8}{2n+1}\right) - \sum_{n=1}^{\infty}\left(\frac{2}{2n-1} - \frac{2}{2n+3}\right) - \sum_{n=1}^{\infty}\frac{8}{(2n+3)^2}\right]\]\[ = \frac{128}{9} - \left[8 - \frac{8}{3} - \left(\pi^2 - \frac{80}{9}\right)\right] = \pi^2 \]

QED– Ariel Gershon · 1 year, 4 months agoLog in to reply

Careful there, you need to justify why you can split an infinite sum into two or more infinite sums. Hint: Prove that all these infinite sums are absolutely convergent.

Otherwise, neat solution! =D =D – Pi Han Goh · 1 year, 4 months ago

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– Ariel Gershon · 1 year, 4 months ago

Good point. I did prove that each of the 3 sums converge, and since all the terms in each one are positive, they are absolutely convergent. :)Log in to reply

– Joel Yip · 1 year, 4 months ago

i totally agreeLog in to reply