Studying Mathematics Thread

[polymath257]

OK, now try

(1^2 + 2^2 + 3^2 +...+n^2)/n^3

and

(1^3 + 2^3 + 3^3 +...+n^3 )/n^4.

Any guesses for a generalization?

After some scrawling and trying to remember old methods, I got

Sum(n^2) = (n/6)(2n^2 + 3n + 1)

Giving limit when divided by n^3 as 1/3.

If I have the energy I'll do n^4 later, but I guess it might be 1/4!

Good guess (assuming that isn't 4 factorial)! Now the challenge: prove it.

Hint: Riemann