Ratio test

For nonzero terms, if

lim sup_{n \to \infty} \frac{a _{n + 1}}{a _{n}} < 1,

then

\sum a_{n}

converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.

Table of contents
- Ratio limit implies root limit Ratio test
- Factorial-over-power series Ratio test

Ratio limit implies root limit

For a positive sequence, if

a_{n + 1} / a_{n} \to L \geq 0

, then

a_{n}^{1/ n} \to L .

For

L > 0

this follows by taking logarithms and applying Cesàro averaging; the case

L = 0

follows from an eventual geometric upper bound.

Factorial-over-power series

For

a_{n} = n! / n^{n}

,

\frac{a _{n + 1}}{a _{n}} = (\frac{n}{n + 1})^{n} \to e^{- 1} .

Thus

\sum n! / n^{n}

converges and

n / (n!)^{1/ n} \to e

.

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The Ratio Test is a method in mathematical analysis, particularly useful for determining the convergence or divergence of infinite series. It is often used for series whose terms involve factorials, exponentials, or other functions where the terms can grow rapidly. ### Statement of the Ratio Test Let \( \{a_n\} \) be a sequence of positive terms.

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