1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.