1. Pi 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 ...

2. Originally Posted by DivideBy0
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 ...

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

3. 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.

4. Originally Posted by DivideBy0
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.
The n-th even number (counting 2 as the first) is 2n, and:

1/(2n)^2 = (1/2^2)(1/n^2) = (1/4)(1/n^2)

RonL