I am trying to solve the following complex form. I will really appreciate the way to solve it. I have tried and failed myself. My results do not even qualify to be mention here.

\begin{equation}

P = e^{-ne^{- \sigma \lambda t}}

\end{equation}

\begin{equation}

E(t) = \int_{0}^{+\infty} t \frac{\partial P}{\partial t}\ dt

\end{equation}

This is part of the paper I am trying to solve.

The solution that the author has given is

\begin{equation}

E(t) = \frac{1}{\sigma \lambda} \sum^n_{k=1} (-1)^{k+1} {{n} \choose {k}} \frac{1}{k}

\end{equation}