Difference between revisions of "D'Alembert criterion (convergence of series)"
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| − | + | {{MSC|40A05}} | |
| + | {{TEX|done}} | ||
| − | + | An elementary criterion to test the convergence of a series of real numbers, established by [[DAlembert|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 | |
| − | + | \[ | |
| − | + | \liminf_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 1 < \limsup_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = 2\, . | |
| − | + | \] | |
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====References==== | ====References==== | ||
| − | + | * {{Ref|Ru}} W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) {{ZBL|0346.26002}} | |
Latest revision as of 11:28, 22 March 2023
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
D'Alembert criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_criterion_(convergence_of_series)&oldid=14634