In the 18th Century, Euler proved the remarkable fact that

1/1^2 + 1/2^2 + 1/3^2+ 1/4^2 + 1/5^2 ... = pi^2/6

Use this to determine the value of

1/1^2 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 ...

Please help me on this... thanks.

Printable View

- April 26th 2007, 10:45 PMDivideBy0Pi Identity
**In the 18th Century, Euler proved the remarkable fact that**

**1/1^2 + 1/2^2 + 1/3^2+ 1/4^2 + 1/5^2 ... = pi^2/6**

**Use this to determine the value of**

**1/1^2 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 ...**

Please help me on this... thanks. - April 26th 2007, 11:24 PMCaptainBlack
1/1^2 + 1/2^2 + 1/3^2+ 1/4^2 + 1/5^2 ... =

....[1/1^2 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 ...] +

.............[1/2^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/10^2 ...] =

....[1/1^2 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 ...] +

.............. [1/[1/4][1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 ...]

So we have:

1/1^2 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 ... = [3/4][1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 ...]

................. = [3/4][pi^2/6] = pi^2/8

RonL - April 27th 2007, 12:23 AMDivideBy0
I don't understand the step where you take a quarter of the reciprocated even squares.

How did you come to the conclusion that

[1/2^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/10^2] = 1/4[1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2]

Can you take maybe a smaller example to show me how this works? Thanks. - April 27th 2007, 02:32 AMCaptainBlack