An infinite summation involving binomial coefficient The question is to find out the value of $$\sum_{n=k}^{\infty} P^n \binom nk (1/2)^n$$
I tried to break down the binomial coefficient and bring it in form of some known sequence but could not get anything out of it.Please help me in this regard.Thanks.

$$\begin{align}
\sum_{n=k}^{\infty}P^n{n\choose k}\left({1\over 2}\right)^n &=  {P^k\over 2^kk!}\sum_{n=k}^\infty n(n-1)\dots(n-k+1)\left({P\over 2}\right)^{n-k}
\\&= {P^k\over 2^kk!}f^{(k)}({P\over 2})
\end{align}$$
Where $f : x\mapsto \sum_{n=0}^\infty x^n = {1\over 1-x}$
Thus $$\sum_{n=k}^{\infty}P^n{n\choose k}\left({1\over 2}\right)^n = {P^k\over 2^kk!}{k!\over (1-P/2)^{k+1}} = {P^k\over 2^k(1-P/2)^{k+1}}$$
