## Summary of Infinite Series

### Essential Concepts

• Given the infinite series

$\displaystyle\sum _{n=1}^{\infty }{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots$

and the corresponding sequence of partial sums $\left\{{S}_{k}\right\}$ where

${S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}$,

the series converges if and only if the sequence $\left\{{S}_{k}\right\}$ converges.

• The geometric series $\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}$ converges if $|r|<1$ and diverges if $|r|\ge 1$. For $|r|<1$,

$\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}$.
• The harmonic series

$\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots$

diverges.

• A series of the form $\displaystyle\sum _{n=1}^{\infty }\left[{b}_{n}-{b}_{n+1}\right]=\left[{b}_{1}-{b}_{2}\right]+\left[{b}_{2}-{b}_{3}\right]+\left[{b}_{3}-{b}_{4}\right]+\cdots +\left[{b}_{n}-{b}_{n+1}\right]+\cdots$

is a telescoping series. The $k\text{th}$ partial sum of this series is given by ${S}_{k}={b}_{1}-{b}_{k+1}$. The series will converge if and only if $\underset{k\to \infty }{\text{lim}}{b}_{k+1}$ exists. In that case,

$\displaystyle\sum _{n=1}^{\infty }\left[{b}_{n}-{b}_{n+1}\right]={b}_{1}-\underset{k\to \infty }{\text{lim}}\left({b}_{k+1}\right)$.

## Key Equations

• Harmonic series

$\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$
• Sum of a geometric series

$\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}\text{ for }|r|<1$

## Glossary

convergence of a series
a series converges if the sequence of partial sums for that series converges
divergence of a series
a series diverges if the sequence of partial sums for that series diverges
geometric series
a geometric series is a series that can be written in the form

$\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\cdots$
harmonic series
the harmonic series takes the form

$\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots$
infinite series
an infinite series is an expression of the form

${a}_{1}+{a}_{2}+{a}_{3}+\cdots =\displaystyle\sum _{n=1}^{\infty }{a}_{n}$
partial sum
the $k\text{th}$ partial sum of the infinite series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ is the finite sum

${S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}$
telescoping series
a telescoping series is one in which most of the terms cancel in each of the partial sums