Show that each series is a telescoping series. For each...

Show that each series is a telescoping series. For each series, provide the general term $S_{n}$ in its sequence of partial sums and find the sum of the series if it converges.
$$
\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{k+2}\right)
$$

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Key Concepts

Telescoping Series

A telescoping series is a series in which many terms cancel when the partial sums are computed, leaving only a few terms that determine the overall sum. This cancellation results from the series being expressed as the difference of consecutive or related terms, which simplifies the process of summing the series.

Partial Sums

Partial sums are formed by adding the first n terms of a series and studying the behavior as n increases. They are essential for analyzing the series because convergence is determined by whether the sequence of partial sums approaches a finite limit.

Series Convergence

Series convergence addresses whether the sum of an infinite series approaches a specific value. By examining the limit of the partial sums, one can determine if the series converges to a finite sum, diverges, or oscillates without settling on any value.

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