You may also like

problem icon


A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

problem icon

The Root of the Problem

Find the sum of this series of surds.

problem icon

Squarely in the Middle

Weekly Problem 20 - 2013
Can you calculate the answer to a large sum?

Sums of Powers - A Festive Story

Stage: 3 and 4
Article by Theo Drane

Published November 2006,December 2006,February 2011.

The general case:

\begin{eqnarray} 1^m +2^m + \dots + n^m \\ = \left.(1^me^t + 2^me^{2t} + \dots n^me^{nt})\right|_{t=0}\\ =\left. \frac{d^m}{dt^m}\left(1 +e^t + e^{2t} + \dots + e^{nt}\right) \right|_{t=0}\\ =\left. \frac{d^{m+1}}{dt^{m+1}}\frac{t(e^{(n+1)t} - 1)}{e^t-1} \right|_{t=0}\\ =\frac{1}{m+1}\left( \left. \frac{d^{m+1}}{dt^{m+1}}\frac{te^{(n+1)t}}{e^t-1}\right|_{t=0} -\left.\frac{d^{m+1}}{dt^{m+1}}\frac{t}{e^t-1}\right|_{t=0}\right) \\ =\frac{1}{m+1}\left( \left. \frac{d^{m+1}}{dt^{m+1}}\sum_{k=0}^{\infty}B_k(n+1)\frac{t^k}{k!}\right|_{t=0} -\left.\frac{d^{m+1}}{dt^{m+1}}\sum_{k=0}^{\infty}B_k\frac{t^k}{k!}\right|_{t=0}\right) \\ =\frac{B_{m+1}(n+1) - B_{m+1}}{m+1}\\ \end{eqnarray}

Where $B_n(x)$ is the Bernoulli polynomial and $B_n$ are the Bernoulli numbers

Conclude that $$1^m + 2^m + \dots + n^m = \frac{B_{m+1}(n+1) - B_{m+1}}{m+1}$$ Amongst other things, all you have to do now is find out what on earth is a Bernoulli polynomial!!