View source for D'Alembert criterion (convergence of series)

{{MSC|40A05}}
{{TEX|done}}

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|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====
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|valign="top"|{{Ref|Ru}}||  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)
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