What is another infinite series summation for Pi^2/6 besides 1/n^2?

1. Jan 13, 2014

mesa

So the title pretty much says it all, what other infinite series summations do we have for Pi^2/6 besides,

$$\sum_{n=1}^{\infty} 1/n^2$$

***EDIT*** I should also include,

$$\sum_{n=1}^{\infty} 2(-1)^(n+1)/n^2$$
$$\sum_{n=1}^{\infty} 4/(2n)^2$$
etc. etc.

A unique form outside of the 1/n^2 family.

Last edited: Jan 13, 2014
2. Jan 14, 2014

dipole

http://www.wolframalpha.com/input/?i=Pi^2%2F6

3. Jan 14, 2014

Office_Shredder

Staff Emeritus
4. Jan 14, 2014

mesa

Interesting, all of their series still have a 'squared' term for the denominator in some form or another. Do you know of any outside of the 1/n^2 family?

Either way thanks for the link!

That thread was about finding infinite series summations for 1/5 and 1/7 which eventually led to a 'general solution' for all '1/k' fractions for infinite series. This thread is specifically for Pi^2/6 and finding infinite series that are not of the 1/n^2 family.

Hope this helps.

5. Jan 14, 2014

Office_Shredder

Staff Emeritus
Take any series that equals 1/5 and multiply it by $\frac{ 5 \pi^2}{6}$.

6. Jan 14, 2014

mesa

Yes, that would do it too...
Do you have anything else besides Boreks standard answer on these things? :)

On a more serious note (sort of), deriving these new series is a blast! I have not encountered any other subject in mathematics that has been more fun!

Last edited: Jan 14, 2014