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)