# D'Alembert criterion (convergence of series)

2020 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
\[
\liminf_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 1 < \limsup_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 2\, .
\]

#### References

- [Ru] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) Zbl 0346.26002

**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=53077