Find $\lim\limits_{n \to +\infty} np^n\sum\limits_{k=n}^{\infty} \frac{1}{k} \binom{k-1}{n-1}(1-p)^{k-n}$.

Question

This is very likely to be a probabilitic problem，just noting that

$$E[1/X]=\sum_{k=n}^{\infty} \frac{1}{k}\binom{k-1}{n-1}p^n(1-p)^{k-n},$$ where $X$ is the random variable of Negative-Binomial Distribution. Thus the present limit is the very $$\lim_{n \to \infty} nE[1/X].$$ How to go on from here? Can it be solved just by calculus without probability?