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#

I hope that this was helpful.

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Answer 2

Emilia Aguilar

A collapsing infinite series is a type of infinite series where most terms cancel each other out, leaving only a finite number of non-zero terms. As more terms are added, the partial sums of the series converge to a finite value. This convergence occurs because the remaining non-zero terms eventually become insignificant compared to the overall behavior of the series. The term "collapsing" refers to the reduction or cancellation of terms during the process of summation, resulting in a simplified and convergent series.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.