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.
Best Answer
Your derivation of $a_n$ using $S_n - S_{n-1}$ is only valid for $n > 1$, i.e. $S_0$ is not defined, or rather, it is defined to be zero, but if you substitute zero in $S_n = \dfrac {n-1}{n+1}$ you get $-1$, which is the cause of the disparity.
We hence need to define $a_1$ separately: In fact $a_1 = S_1 = 0$. Hence:
$$S = \sum_{n=1}^\infty a_n = \sum_{n=\color{red}2}^\infty\frac 2{n(n+1)} = 1$$