TITLE: Why does this Infinite Series have contradictory convergences?
QUESTION [14 upvotes]: In my Calculus II homework, I encountered the following exercise:

If the $n$th partial sum of a series $\sum_{n=1}^\infty a_n$ is
$$s_n = \frac {n-1}{n+1}$$
find  $a_n$ and $\sum_{n=1}^\infty a_n$.

I solved the exercise this way: ( I took $S_n=(n-1)/(n+1)$ to be equation (1))

Equations (2) and (3) answer the exercise's questions. However, when I tried to corroborate my answer in equation (3) by using the found $a_n$ in equation (2), I encountered an inconsistency that I haven't yet been able to harmonize. This is what I tried to do:

Equations (3) and (4) should have yielded the same answer but, this disparity I have been so far unable to harmonize. Your kind comments on what to do will be greatly appreciated.

REPLY [2 votes]: This is not an answer, just a small addition to the post of player3236:
As already in the post of player3236 mentioned $s_n$ is only definied for all $n\in\mathbb{N}$. So you can't calculate $s_n-s_{n-1}$ for all $n\in\mathbb{N}$. But you can calculate
$$s_{n+1}-s_n=a_{n+1}$$
without changing the range of n. By a similar calculation as you did above you will receive $a_{n+1}=\frac{2}{(n+1)(n+2)}$ for all $n\in\mathbb{N}$. Now you still have to find $a_1$, which can either be done directly by the given formular $a_1=s_1=\frac{1-1}{1+1}=0$ or by using the formular for $a_{n+1}$
$$s_2=\frac{1}{3}=\frac{2}{(1+1)(1+2)}=a_2
$$
and thus $a_1=0$ since $s_2=a_2+a_1$.