Given that:
(1)-4 + (3)-4 + (5)-4 + ... = π4/96
Then, find the value of:
(1)-4 + (2)-4 + (3)-4 + ...
Lets start with (1/1)^4 + (1/2)^4 + (1/3)^3 + (1/4)^4 + ... = S
Split the terms into odd and even:
[(1/1)^4 + (1/3)^4 + ...] + [(1/2)^3 + (1/4)^4 + ... ] = S
Pull out a common factor of (1/2)^4 from the right sum:
[(1/1)^4 + (1/3)^4 + ...] + (1/2)^4 * [(1/1)^3 + (1/2)^4 + ... ] = S
Now the left sum is the given sum and the right sum is the first sum. Then after substituting:
pi^4/96 + (1/2)^4 * S = S
Now its simple to solve for S=pi^4/90.
Solution
