I'm trying to find the sequence $(a_n)_{n\in\mathbb{N_+}}$ (in terms of $x$) from the partial sum of the series: $\sum_{n=1}^\infty a_n$, where the partial sum is: $$s_k=\sum_{n=1}^k a_n=\frac{(1+x)^k-x^k}{(1+x)^k}, x>0, k\in\mathbb{N}_+.$$ I know how to find: $\sum_{n=1}^\infty a_n$, i.e., $\lim_{n\to \infty}s_k=1$ (I don't know if that helps), but I'm stuck on how to find $a_n$. Thanks in advance.


Hint. One has $$ s_n-s_{n-1}=a_n,\qquad n=2,3,4,\cdots, $$ with $s_1=a_1$.

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  • $\begingroup$ Perhaps move those subscripts around a bit and this will be perfect. $\endgroup$ – Simply Beautiful Art Dec 3 '16 at 19:35
  • $\begingroup$ @SimpleArt Edited, thanks! $\endgroup$ – Olivier Oloa Dec 3 '16 at 19:36

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