Summary of Infinite Series

Given the infinite series

[latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots[/latex]

and the corresponding sequence of partial sums [latex]\left\{{S}_{k}\right\}[/latex] where

[latex]{S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}[/latex],

the series converges if and only if the sequence [latex]\left\{{S}_{k}\right\}[/latex] converges.
The geometric series [latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}[/latex] converges if [latex]|r|<1[/latex] and diverges if [latex]|r|\ge 1[/latex]. For [latex]|r|<1[/latex],

[latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}[/latex].
The harmonic series

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots[/latex]

diverges.
A series of the form [latex]\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[/latex]

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

[latex]\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)[/latex].

Key Equations

Harmonic series

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots[/latex]
Sum of a geometric series

[latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}\text{ for }|r|<1[/latex]

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

[latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\cdots[/latex]

harmonic series: the harmonic series takes the form

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots[/latex]

infinite series: an infinite series is an expression of the form

[latex]{a}_{1}+{a}_{2}+{a}_{3}+\cdots =\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex]

partial sum: the [latex]k\text{th}[/latex] partial sum of the infinite series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is the finite sum

[latex]{S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}[/latex]

telescoping series: a telescoping series is one in which most of the terms cancel in each of the partial sums