Summary of the Divergence and Integral Tests

Essential Concepts

  • If [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0[/latex], then the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges.
  • If [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}=0[/latex], the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] may converge or diverge.
  • If [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is a series with positive terms [latex]{a}_{n}[/latex] and [latex]f[/latex] is a continuous, decreasing function such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], then

    [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{and}{\displaystyle\int }_{1}^{\infty }f\left(x\right)dx[/latex]



    either both converge or both diverge. Furthermore, if [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges, then the [latex]N\text{th}[/latex] partial sum approximation [latex]{S}_{N}[/latex] is accurate up to an error [latex]{R}_{N}[/latex] where [latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex].

  • The p-series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}[/latex] converges if [latex]p>1[/latex] and diverges if [latex]p\le 1[/latex].

Key Equations

  • Divergence test

    [latex]\text{If }{a}_{n}\nrightarrow 0\text{ as }n\to \infty ,\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{ diverges}[/latex].
  • p-series

    [latex]{\displaystyle\sum _{n=1}^{\infty}} \dfrac{1}{n^{p}} \bigg\{ \begin{array}{l}\text{ converges if }p>1\\ \text{ diverges if }p\le 1\end{array}[/latex]
  • Remainder estimate from the integral test

    [latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex]

Glossary

divergence test
if [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0[/latex], then the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges
integral test
for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms [latex]{a}_{n}[/latex], if there exists a continuous, decreasing function [latex]f[/latex] such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], then

[latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{ and }{\displaystyle\int }_{1}^{\infty }f\left(x\right)dx[/latex]
either both converge or both diverge
p-series
a series of the form [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}[/latex]
remainder estimate
for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms [latex]{a}_{n}[/latex] and a continuous, decreasing function [latex]f[/latex] such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], the remainder [latex]{R}_{N}=\displaystyle\sum _{n=1}^{\infty }{a}_{n}-\displaystyle\sum _{n=1}^{N}{a}_{n}[/latex] satisfies the following estimate:

[latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex]