Difference between revisions of "Ermakov convergence criterion"
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| − | A criterion for the convergence of a series $\sum_n f(n)$, where $f:[1, \infty[\to [0, \infty[$ is a monotone decreasing function, established by | + | A criterion for the convergence of a series $\sum_n f(n)$, where $f:[1, \infty[\to [0, \infty[$ is a monotone decreasing function, established by V.P. Ermakov in {{Cite|Er}}. |
| − | Let | + | Let $f(x)$ be a positive decreasing function for $x \ge 1$. If there is $\lambda< 1$ such that |
\[ | \[ | ||
\frac{e^x f(e^x)}{f(x)} < \lambda | \frac{e^x f(e^x)}{f(x)} < \lambda | ||
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\frac{e^x f(e^x)}{f(x)}\geq 1 | \frac{e^x f(e^x)}{f(x)}\geq 1 | ||
\] | \] | ||
| − | for all sufficiently large $x$, then the series diverges. In particular the convergence or divergence of the series can be decided | + | for all sufficiently large $x$, then the series diverges. In particular the convergence or divergence of the series can be decided if the limit |
\[ | \[ | ||
\lim_{x\to\infty} \frac{e^x f(e^x)}{f(x)} | \lim_{x\to\infty} \frac{e^x f(e^x)}{f(x)} | ||
Latest revision as of 20:55, 23 December 2014
2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]
A criterion for the convergence of a series $\sum_n f(n)$, where $f:[1, \infty[\to [0, \infty[$ is a monotone decreasing function, established by V.P. Ermakov in [Er].
Let $f(x)$ be a positive decreasing function for $x \ge 1$. If there is $\lambda< 1$ such that \[ \frac{e^x f(e^x)}{f(x)} < \lambda \] for sufficiently large $x$, then the series $\sum_n f(n)$ converges. If instead \[ \frac{e^x f(e^x)}{f(x)}\geq 1 \] for all sufficiently large $x$, then the series diverges. In particular the convergence or divergence of the series can be decided if the limit \[ \lim_{x\to\infty} \frac{e^x f(e^x)}{f(x)} \] exists and differs from 1.
Ermakov's criterion can be derived from the integral test.
References
| [Br] | T.J. Bromwich, "An introduction to the theory of infinite series" , Macmillan (1947) |
| [Er] | V.P. Ermakov, "A new criterion for convergence and divergence of infinite series of constant sign" , Kiev (1872) (In Russian) |
Ermakov convergence criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ermakov_convergence_criterion&oldid=30931