Ratio Test

Summary

The ratio test examines $L = lim | a_{n + 1} / a_{n} |$ . Absolute convergence holds if $L < 1$ ; divergence if $L > 1$ ; the test is inconclusive if $L = 1$ .

Prerequisites

Absolute Convergence, factorials and exponentials helpful

Theorem

For $\sum a_{n}$ , if

L = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} |

exists, then:

$L < 1$ : absolute convergence;
$L > 1$ or $L = \infty$ : divergence;
$L = 1$ : inconclusive (e.g. all $p$ -series).

Worked Example

$\sum (1 / 2)^{n}$ : $L = 1 / 2 < 1$ , absolute convergence.

$\sum n! / (2 n)!$ :

| \frac{a_{n + 1}}{a_{n}} | = \frac{n + 1}{(2 n + 2) (2 n + 1)} \to 0 < 1,

so absolute convergence.

For $\sum x^{n} / n$ , $L = | x |$ : absolute convergence when $| x | < 1$ , divergence when $| x | > 1$ ; endpoints need separate checks.

Common Mistakes

Concluding anything from $L = 1$ .
Dropping absolute values for alternating terms when testing absolute convergence.

Connections

Root Test, Power Series (radius via ratios of coefficients)

References

The ratio test is in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Section 5.6, https://openstax.org/details/books/calculus-volume-2 ↩︎