Question

What is a telescoping infinite series?

Answer 1

Here is an example of a telescoping series
`sum_{n=1}^infty(1/n-1/{n+1})`
`=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+cdots`
As you can see above, terms are shifted with some overlapping terms, which reminds us of a telescope. In order to find the sum, we will its partial sum `S_n` first.
`S_n=(1/1-1/2)+(1/2-1/3)+cdots+(1/n-1/{n+1})`
by cancelling the overlapping terms,
`=1-1/{n+1}`
Hence, the sume of the infinite series can be found by
`sum_{n=1}^infty(1/n-1/{n+1})=lim_{n to infty}S_n=lim_{n to infty}(1-1/{n+1})=1`