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Prove that $\sum\limits_{k=1}^{\infty} a_k$ is convergent if $\lim\limits_{k\to\infty} k(1 - \frac{a_{k+1}}{a_k})= \alpha$, where $\alpha > 1$ and $ k = 1, 2, ...$

This is my progress so far:

$ k(1-\frac{a_{k+1}}{a_k}) = \alpha $

$\iff a_{k+1} = (1-\frac{a}{k})a_k$

By this we can see that $a_k$ is decreasing.

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    $\begingroup$ This is Raabe's test. $\endgroup$ – Lord Shark the Unknown May 27 '17 at 11:23
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    $\begingroup$ Pick $b$ in $(1,a)$ and show recursively that there exists $c$ and $n_0$ such that, for every $n\geqslant n_0$, $$a_n\leqslant cn^{-b}$$ To do that, it will help you to show that, for every $k$ large enough, $$\left(1-\frac{b}k\right)\frac1{k^b}\leqslant\frac1{(k+1)^b}$$ $\endgroup$ – Did May 27 '17 at 11:31

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