# Vanishing Condition and nth partial sum question

If the nth partial Sum of the series $\Sigma a_n$ is $s_n = \frac {n-1}{n+1}$ find $a_n$ and $\Sigma a_n$.

So from my understanding I need to use the vanishing condition which is a corollary of the div test (convergence/divergence tests) for series. I am really confused with this one, its a challenge question, but I just am not sure.

• A telescoping series will work fine. Notice $S_n = 1-\dfrac{2}{n+1}$. I'm sure you have encountered similar. – David Peterson Jul 23 '17 at 21:52
• @Rex: Hope this formatting is reflective of what you wanted. Roll it back if it is not. – gary Jul 23 '17 at 21:52
• @gary, perfect formatting, thank you – Rex Jul 23 '17 at 21:55
• Hint: $\,s_n=a_n+s_{n-1}\,$, so $a_n=s_n-s_{n-1}\,$. – dxiv Jul 23 '17 at 23:58

$s_{n}=s_{n-1}+a_n\to a_n=s_n-s_{n-1}=\dfrac{n-1}{n+1}-\dfrac{n-1-1}{n-1+1}=\dfrac{n-1}{n+1}-\dfrac{n-2}{n}$
$a_n=\dfrac{2}{n^2+n}$
$s=\mathop {\lim }\limits_{n \to \infty } \, \dfrac{n-1}{n+1}=1$