Give $x \in (0,0.5)$, how can we compute the asymptotic result of $\sum\_{k=n}^{\infty} {k+n \choose n} x^{k}$ as $n \rightarrow \infty$? Thanks. 



An asymptotic question
Concerning the case $x\rightarrow \frac{1}{2}$ we have :  

$f(1/2,n)= 2^n+\binom{2n}{n} 2^{-n}$ (nearly $2^n (1+1/\sqrt{\pi n}))$  

found at page 247 of *Concrete Mathematics* Graham, Knuth, and Patashnik. See too equation (5.20) and the discussion in ['partial sum involving factorials'](https://math.stackexchange.com/questions/95236/partial-sum-involving-factorials) of stackexchange.  

(this should have been a comment anyway...)


