I'm trying to calculate the expectation of a random variable and I've ended up with the following summation that I need to evaluate. I've tried a number of things including summation by parts, etc, but I can't find a way of solving it. Is there a closed form solution?

$$ \sum_{k=0}^{n} \binom nk\frac {p^k (1-p)^{n-k}}{a+k} $$

Here, $a>0$ is an arbitrary integer parameter.


  • $\begingroup$ it leads to a hypergeometric function $\endgroup$ – Dr. Sonnhard Graubner Dec 15 '16 at 14:37
  • $\begingroup$ @Dr.SonnhardGraubner How would I go about determining the parameters of the hypergeometric function that represents it? $\endgroup$ – N. Price Dec 15 '16 at 15:52

It looks like a dot product between a row vector the elements of which are the probability of some events and a column vector the elements of which are 1/(a+k).


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