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.
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
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.