Infinite Series

Home

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 $\displaystyle\sum_{n=2}^\infty \frac{1}{n(n-1)}=1$.
.

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.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

Models of Growth

Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

Conic sections

Conic Sections in Polar Coordinates

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Functions of 2 and 3 variables

Partial Derivatives

Differentiability and the Chain Rule

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Double integrals in polar coordinates

Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables

Telescoping Sums