Telescoping Sums

A telescoping sum is a sum of differences.  We look at a general example.  If $a_n = f(n)-f(n+1)$, then $$\sum_{n=1}^\infty a_n =\sum_{n=1}^\infty \left(f(n)-f(n+1)\right)= \left(f(1)-f(2)\right)+\left(f(2)-f(3)\right)+\left(f(3)-f(4)\right) + \cdots.$$DO:  Carefully find the $n^{th}$ partial sum of this series.  It is a telescoping sum.  $\displaystyle s_n=\sum_{i=1}^n \left(f(i)-f(i+1)\right) = \left(f(1)-f(2)\right)+\left(f(2)-f(3)\right) + \cdots+(f(n-1)-f(n))+(f(n)-f(n+1))$.  You can cancel many terms (Do this.) to be left with only the two terms $s_n=f(1)-f(n+1)$.  Thus we can compute the convergence/divergence of the series, and if it converges, we can find its value: $$\sum_{n=1}^\infty \left(f(n)-f(n+1)\right)=\lim_{n\to\infty}s_n=\lim_{n\to\infty}(f(1)-f(n+1))=f(1)-\lim_{n\to\infty}f(n+1).$$A series with telescoping partial sums is one of the rare series with which we can compute the value of the series by using the definition of a series as the limit of its partial sums.

Example: $\displaystyle\sum_{n=2}^\infty \frac{1}{n(n-1)} = \sum_{n=2}^\infty\left( \frac{1}{n-1}-\frac{1}{n}\right)$.  DO:  Check this equality by using partial fraction decomposition on $\displaystyle\frac{1}{n(n-1)}$, then write out the $n^{th}$ partial sum of this series and cancel terms.



$\displaystyle s_n=\sum_{i=2}^n\left( \frac{1}{i-1}-\frac{1}{i}\right)=\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\cdots+\left(\frac{1}{n-2}-\frac{1}{n-1}\right)+\left(\frac{1}{n-1}-\frac{1}{n}\right)$
                                   $\displaystyle=1-\frac{1}{n}$
DO:  find the value of the series by computing the limit of the partial sums.


$\displaystyle\lim_{n\to\infty}s_n=\lim_{n\to\infty}\left(1-\frac{1}{n}\right)=1-0=1$.  So $\displaystyle\sum_{n=2}^\infty \frac{1}{n(n-1)}=1$.
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This video will play with these ideas in a more abstract way, and will tie them to integrals, derivatives, and the FTC.  Depending upon your instructor, the information in the video, while interesting, is optional.