Exercises

These problems are all related, and are all pretty mechanical. You may wish to do a few of them just to exercise your algebra and a mechanical application of induction. Some involve a lot of grinding--they're mechanical, not necessarily easy!

Each series below has $n$ terms:

\begin{eqnarray*} 0^1 + 1^1 + 2^1 + 3^1 + \cdots + (n-1)^1 &=& {n^2\over 2} - {n... ...2} +{2n^7\over 3} - {7n^5\over 15} + {2n^3\over 9} - {n\over 30} \end{eqnarray*}