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#

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Answer 2

Charles Anctil

A telescoping infinite series is a series where most of the terms cancel each other out, leaving only a finite number of terms to be summed. This cancellation typically occurs in a pattern such that consecutive terms partially or completely cancel each other, causing the series to "collapse" or "telescope" down to a finite sum. Telescoping series are often used in mathematics to simplify the computation of infinite sums.

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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.