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

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mesa
#1
Jan13-14, 11:35 PM
P: 559
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.
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dipole
#2
Jan14-14, 01:44 PM
P: 444
http://www.wolframalpha.com/input/?i=Pi^2%2F6
Office_Shredder
#3
Jan14-14, 01:50 PM
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How is this thread different than the one you posted before?

http://www.physicsforums.com/showthread.php?t=730791

mesa
#4
Jan14-14, 07:01 PM
P: 559
What is another infinite series summation for Pi^2/6 besides 1/n^2?

Quote Quote by dipole View Post
http://www.wolframalpha.com/input/?i=Pi^2%2F6
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!

Quote Quote by Office_Shredder View Post
How is this thread different than the one you posted before?

http://www.physicsforums.com/showthread.php?t=730791
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.
Office_Shredder
#5
Jan14-14, 07:37 PM
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Quote Quote by mesa View Post
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.
Take any series that equals 1/5 and multiply it by [itex] \frac{ 5 \pi^2}{6} [/itex].
mesa
#6
Jan14-14, 08:25 PM
P: 559
Quote Quote by Office_Shredder View Post
Take any series that equals 1/5 and multiply it by [itex] \frac{ 5 \pi^2}{6} [/itex].
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!


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