How do you find the n-th partial sum of an infinite series?

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Answer 1 · 2014-09-19T14:27:12

The nth partial sum $S_n$ of a series $sum_{k=1}^infty a_k$ is
$S_n=sum_{k=1}^n=a_1+a_2+a_3+cdots+a_n$

Let us find $S_n$ for the telescoping series

$sum_{k=1}^infty(1/k-1/{k+1})$ .

The partial sum is

$S_n=(1/1-1/2)+(1/2-1/3)+cdots+(1/n-1/{n+1})$

by regrouping,

$=1+(-1/2+1/2)+cdots+(1/n-1/n)-1/{n+1}$

$=1+0+cdots+0-1/{n+1}$

$=1-1/{n+1}=n/{n+1}$

Hence, $S_n=n/{n+1}$ .