# D'Alembert criterion (convergence of series)

2010 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

An elementary criterion to test the convergence of a series of real numbers, established by J. d'Alembert in 1768, and which is also known as ratio test. Consider such a series $\sum_n a_n$ and assume that $a_n \neq 0$.

• If

$\limsup_{n\to \infty} \frac{|a_{n+1}|}{|a_n|} < 1$ then the series converges absolutely

• If

$\liminf_{n\to \infty} \frac{|a_{n+1}|}{|a_n|} > 1\, ,$ then the series diverges.

None of the conclusions can be extended to the equality case. In particular for both the harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ (which diverges) and the series $\sum_{n=1}^\infty \frac{1}{n^2}\,$ (which converges) one has $\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = 1\, .$ A slight modification of these two examples provide also the following conclusions

Example 1 Let $\{N_k\}$ be an increasing sequence of natural numbers such that $\sum_{n=N_k+1}^{N_{k+1}} \frac{1}{n} \geq 2^k\,$ and set $a_n := \frac{1}{2^k n} \qquad \mbox{ for } N_k+1\leq n \leq N_{k+1}\, .$ It is easy to check that the series $\sum a_n$ diverges and $\liminf_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = \frac{1}{2} < 1 = \limsup_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}\, .$

Example 2 Let $\{N_k\}$ be an increasing sequence of natural numbers such that $\sum_{n>N_k}^\infty \frac{1}{n^2} < 3^{-n}\,$ and set $a_n = \frac{2^k}{n^2} \qquad \mbox{ for } N_k+1\leq n \leq N_{k+1}\, .$ It is easy to check that the series $\sum a_n$ converges and $\liminf_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 1 < \limsup_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 2\, .$

How to Cite This Entry:
D'Alembert criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_criterion_(convergence_of_series)&oldid=30909
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article