Science:Infinite Series Module/Units/Unit 1/1.5 The Telescoping and Harmonic Series/1.5.05 Final Notes on Harmonic and Telescoping Series

A Slightly More General Harmonic Series Form

We were able to show that

${\begin{aligned}\sum _{k=1}^{\infty }{\frac {1}{k}}\end{aligned}}$

is divergent. It can be shown that

${\begin{aligned}\sum _{k=1}^{\infty }{\frac {1}{k-a}}\end{aligned}}$

is also divergent, where $a$ is any real number. Observe that

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \sum_{k=1}^{\infty} \frac{1}{k-a} &= \sum_{k=1}^{a-1} \frac{1}{k-a}+ \sum_{k=a}^{\infty} \frac{1}{k-a} \\ &= \sum_{k=1}^{a-1} \frac{1}{k-a}+ \sum_{k=1}^{\infty} \frac{1}{k}. \end{align}}

The first sum is a constant, while the second sum is the harmonic series, which is divergent.

General Form of a Telescoping Series

One might ask how to identify an infinite telescoping series. It may seem a bit obvious, but for the sake of completeness, a telescoping series could have the form

$\sum _{k=1}^{\infty }{\Big (}{\frac {1}{k+a}}-{\frac {1}{k+b}}{\Big )}$

where integers $a$ and $b$ satisfy $a<b$. Indeed, one could imagine more complicated forms of telescoping series, but for our purposes, this will be sufficient. Students will only need to be familiar with this form of the telescoping series.

Telescoping Series May Require Partial Fractions

Computing sums of telescoping series can involve having able to compute a partial fraction decomposition. If you would like additional practice with telescopic series that use partial fractions, you can find additional examples that involve the telescoping series in the Additional Resources section within this lesson.