Formula for sum of combinations I am trying to find a closed-form formula or, at least, a different and more useful representation for the following sum of combinations:
\begin{equation}
\sum_{i=1}^{n}\frac{n!}{i!(n-i)!}\times\frac{(-1)^i}{i}
\end{equation}

Hint : sub this into the sum
\begin{eqnarray*}
\frac{1}{i}=\int_0^1 x^{i-1} dx.
\end{eqnarray*}
Now invert the order of the integral and the plum. Use the binomial theorem, do the obvious substitution, expand and integrate to get the result stated by Interstellar Probe.
