TITLE: Where is the mistake in this problem?
QUESTION [11 upvotes]: Let $$S = \sum^{\infty}_{n=1} a(n)$$
be an infinite series such that the nth partial sum is given by:
$$S(n) = \frac{n - 1}{n + 1}$$
since $$
a(n)=S(n)-S(n-1)=\frac{2}{n(n+1)} 
$$
Now, $S(1)=a(1)\Leftrightarrow 0=1.$  Where is the mistake?

REPLY [4 votes]: Depending on your definition there are two possible problems:
If S(0) exists:
Then $S(0)$ has to definitions as $S(0)=\sum_{n=1}^0a(n)=0$ and $S(0)=\frac{0-1}{0+1}=-1$
If $S(0)$ doesn't exists: Then $S(1)$ is your first term and $a(1)=S(1)-S(0)$ isn't a valid formula for $a(1)$