Hi, I am stuck at following problem in my research.

Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. $\mathcal{E} _M$ is the expectation operator with respect to $M$. I need to calculate following expectations: \begin{equation} \mathcal{E} _M \left[ \frac{m}{m+1} e^{ - \frac{2^{\frac{m+1}{m}\cdot r } - 1}{\gamma}} \right] \end{equation} and \begin{equation} \mathcal{E} _M \left[ \frac{m}{m+1} \left(e^{ - \frac{2^{\frac{m+1}{m}\cdot r } - 1}{\gamma}}\right)^{m-1} - \frac{m}{m+1} \left(e^{ - \frac{2^{\frac{m+1}{m}\cdot r } - 1}{\gamma}}\right)^{m} \right] \end{equation} The values of $r$ and $\gamma$ are such that the exponential function cannot be approximated.

Can anyone please provide any guidance/reference for how to go about solving/approximating above expectations? Thanks in advance.