# Approximate sum with binomial coefficients

Is there an efficient way to compute

$\sum_{i=1}^n \binom{n}{i} p^i (1-p)^{n-i} / i$

I tried to look at some properties of the binomial coefficient to get rid of the $1/i$, but it seems there is no closed solution. Can I use an approximation?

Ok, I found the normal approximation:

$\frac{1}{\sqrt{2\pi np(1 - p)}}\sum_{i=1}^{n}\exp(\frac{-(i - np)^2}{(2 np(1 - p))})/i$

which is good enough for me.

• Approximation as $n \to \infty$? – Olivier Oloa Jan 3 '18 at 13:50
• No, actually I need the finite sums for different $n$. Thanks for the hint, I will check it. I would also like to understand how to get from the sum to the integral. – Karsten W. Jan 3 '18 at 14:30
• I would aim to work out cumulative distribution of a variation on Poisson distribution obtained via Poisson limit theorem. Will answer if managed. – Lukáš Mrazík Jan 3 '18 at 15:22