Proving Series Equations: A General Method

[glebovg]

How to prove (not by induction)

$1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}$?

What is the general approach for similar series, say, $1^{1}+2^{2}+\ldots+n^{n}$?

 

[sunjin09]

How to prove (not by induction)

$1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}$?

What is the general approach for similar series, say, $1^{1}+2^{2}+\ldots+n^{n}$?

For your first question, the sum is the solution to the difference equation S(n)-S(n-1)=n^2 subject to the initial condition S(1)=1. Since the difference is 2nd order polynomial, the solution is 3rd order polynomial, now you know how to proceed. For your second question, since the difference is n^n, no known simple function of n has such difference, therefore no simple solution.