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Let $S_n$ be a series with binomial coefficients as follows:

$$ S_n = \sum_{k=1}^n \begin{pmatrix} n \\ k \end{pmatrix}\frac 1k\left(-\frac{1}{1-a} \right)^k\left(\frac{a}{1-a} \right)^{n-k},$$ where $0 < a < 1$. My question is: when $n\to +\infty$, does the series $S_n$ converge? And If it converges, can we find the limit $ \underset{n\to +\infty}{lim} S_n$ ? Thank you.

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We have:

$$\begin{eqnarray*} S_n = \frac{1}{(1-a)^n}\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^k}{k}a^{n-k}&=&\frac{1}{(1-a)^n}\sum_{k=1}^{n}\binom{n}{k}a^{n-k}\int_{0}^{-1}x^{k-1}\,dx\\&=&\frac{1}{(1-a)^n}\int_{0}^{-1}\frac{dx}{x}\sum_{k=1}^{n}\binom{n}{k}x^k a^{n-k}\,dx\\&=&\frac{1}{(1-a)^n}\int_{0}^{-1}\frac{(a+x)^n-a^n}{x}\,dx\\&=&\left(\frac{a}{1-a}\right)^n \int_{0}^{-1/a}\frac{(1+x)^n-1}{x}\,dx\end{eqnarray*} $$ and by approximating the last integrand function with $n\cdot \exp\left(\frac{n-1}{2}x\right)$ (Laplace's method) we have that the limit is zero if $a<\frac{1}{2}$, $-2$ if $a=\frac{1}{2}$ and $-\infty$ if $a>\frac{1}{2}$.

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