I am reading a paper and encountering an asymptotic expression: 

$f(R)=\sum\_{i=1}^{R} \binom{R}{i} (-1)^{i+1} \frac{1}{1-p^i}$

Then, the author says $O(f(R))=O\left(1+\frac{p\ln R}{1-p}\right)$. May I know how this asymptotic form is derived?