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}#.