Show that the series ∑n=1^∞ an can be written in the telescoping form ∑n=1^∞[(c-Sn-1)-(c-Sn)] wh

step 1 In this question, we consider the series of the form summation of the AN. For the N gets from 1

Show that the series ∑n=1^∞ an can be written in the...

Show that the series $\sum_{n=1}^{\infty} a_{n}$ can be written in the telescoping form
$\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_{n}\right)\right]$
where $S_{0}=0$ and $S_{n}$ is the $n$ th partial sum.

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Key Concepts

Telescoping Series

A telescoping series is one in which many terms cancel each other when the series is written in expanded form. The typical structure of a telescoping series involves expressing the general term as a difference of two successive terms, leading to a collapse or cancellation of intermediate terms upon summation. This simplifies the evaluation of the series, often reducing the sum to just a few non-canceling terms at the beginning and end of the expanded form.

Partial Sums

Partial sums are the sums of the first n terms of a series and are fundamental in understanding the convergence or divergence of an infinite series. In many problems, the series' behavior is analyzed via its partial sums. They are often denoted by S? and serve as a key tool in expressing and manipulating series, such as when rewriting series in telescoping form.

Series Manipulation

Rewriting series by algebraic manipulation is an important technique in analysis. In this context, the original series is rearranged into a telescoping form by expressing each term as the difference between successive partial sums. This approach simplifies the summation by harnessing cancellation properties, which is a common strategy in handling complex series.

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