The discussion centers on proving the series equation for the sum of squares, specifically the formula \(1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}\). The approach involves solving the difference equation \(S(n)-S(n-1)=n^2\) with the initial condition \(S(1)=1\), leading to a third-order polynomial solution. In contrast, the series \(1^{1}+2^{2}+\ldots+n^{n}\) presents a challenge as its difference \(n^n\) does not yield a known simple function of \(n\).
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glebovg said: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}?