Summary of Infinite Series

Essential Concepts

  • 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]


  • 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