Convergence Tests

Level 1 - Calculus topic page in Series And Approximations.

Principle

Convergence tests give structured ways to decide whether a series converges without finding its exact sum. The correct test depends on the form and sign of the terms.

Notation

\(a_n\)

nth term of a series

\(p\)

power in a p-series

\(r\)

ratio-test limit

The Core Method

Common tests include the term test, comparison tests, \(p\)-series test, ratio test, and alternating series test.

p-series rule

\[\sum_{n=1}^{\infty}\frac{1}{n^p}\text{ converges exactly when }p\gt1\]

For the ratio test, compute \(r=\lim |a_{n+1}/a_n|\). The series converges if \(r<1\), diverges if \(r>1\), and needs another test if \(r=1\).

Worked Cases

Question

Test \(\sum_{n=1}^{\infty}\frac{1}{n!}\) for convergence.

Answer

Let \(a_n=1/n!\). Then \(\left|\frac{a_{n+1}}{a_n}\right|=\frac{1/(n+1)!}{1/n!}=\frac{1}{n+1}\). The limit is \(0\), which is less than \(1\). Therefore the series converges.

Examples

Question

Does \(\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}\) converge?

Answer

This is a \(p\)-series with \(p=3/2\). Since \(3/2>1\), the series converges.

Series

Power Series