Summary of the Divergence and Integral Tests

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

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

either both converge or both diverge. Furthermore, if $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges, then the $N\text{th}$ partial sum approximation ${S}_{N}$ is accurate up to an error ${R}_{N}$ where ${\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx$ .
The p-series $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}$ converges if $p>1$ and diverges if $p\le 1$ .

Key Equations

Divergence test
$\text{If }{a}_{n}\nrightarrow 0\text{ as }n\to \infty ,\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{ diverges}$ .
p-series
${\displaystyle\sum _{n=1}^{\infty}} \dfrac{1}{n^{p}} \bigg\{ \begin{array}{l}\text{ converges if }p>1\\ \text{ diverges if }p\le 1\end{array}$
Remainder estimate from the integral test
${\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx$

divergence test: if $\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0$ , then the series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ diverges

integral test: for a series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ with positive terms ${a}_{n}$ , if there exists a continuous, decreasing function $f$ such that $f\left(n\right)={a}_{n}$ for all positive integers $n$ , then

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

either both converge or both diverge

p-series: a series of the form $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}$

remainder estimate: for a series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ with positive terms ${a}_{n}$ and a continuous, decreasing function $f$ such that $f\left(n\right)={a}_{n}$ for all positive integers $n$ , the remainder ${R}_{N}=\displaystyle\sum _{n=1}^{\infty }{a}_{n}-\displaystyle\sum _{n=1}^{N}{a}_{n}$ satisfies the following estimate:

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