Question

What is a collapsing infinite series?

Answer 1

Here is an example of a collapsing (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 ("collapsing") 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`

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I hope that this was helpful.