The Converges(f, k, m, n) and Converges(s) commands check if the series $s$ converges unconditionally, and return $\mathrm{true}$ if it does, $\mathrm{false}$ if it diverges unconditionally, and $\mathrm{FAIL}$ otherwise. Typically, $n\=\infty$, but alternatively $m\=-\infty$ is also possible.

The return value of $\mathrm{FAIL}$ can indicate either that $s$ neither converges unconditionally nor diverges unconditionally, i.e., the convergence behavior depends on the value of a parameter, or that the command was unable to determine the convergence behavior for some other reason. In the first case, the ConvergenceRadius can be used to find a condition for convergence.

The ConvergenceRadius(f, k, m, n, x) and ConvergenceRadius(s, x) commands determine the radius of convergence $r$ of the series $s$ w.r.t. $x$. The result is returned in the form $\&verbar;x\&verbar;<r$, where $r\in {ℝ}_0^{\&plus;}\cup \left\{\infty \right\}$.  $r$ has the additional property that $s$ diverges when $\&verbar;x\&verbar;\&gt;r$.

If $s$ is not a Taylor, Laurent or Puiseux series, or if the expansion point of $s$ is not $0$, then ConvergenceRadius may return a more general inequality of the form $\&verbar;\cdots \&verbar;<r$, where $r$ is not necessarily the convergence radius.

If $s$ converges unconditionally, then ConvergenceRadius returns $\&verbar;x\&verbar;<\infty$, and if $s$ diverges unconditionally, the result is $\left|x\right|<0$.

All calling sequences may return $\mathrm{FAIL}$ if the convergence condition cannot be determined, or if $m\&equals;-\infty$ and $n\&equals;\infty$.

If both $m$ and $n$ are finite, then Converges returns $\mathrm{true}$, and ConvergenceRadius returns $\&verbar;x\&verbar;<\infty$.

$\mathrm{with}\left(\mathrm{SumTools}:-\mathrm{DefiniteSum}\right)$

$\left[\mathrm{ConvergenceRadius}\&comma;\mathrm{Converges}\&comma;\mathrm{CreativeTelescoping}\&comma;\mathrm{Definite}\&comma;\mathrm{SummableSpace}\&comma;\mathrm{Telescoping}\&comma;\mathrm{pFqToStandardFunctions}\right]$

$\mathrm{Converges}\left(\frac{1}{n}\&comma;n\&comma;1\&comma;\mathrm{\infty }\right)$

$\mathrm{false}$

$\mathrm{s0}≔\mathrm{Sum}\left(\frac{1}{n}\&comma;n=1..\mathrm{\infty }\right)$

$\mathrm{s0}≔\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$

$\mathrm{Converges}\left(\mathrm{s0}\right)$

$\mathrm{false}$

$\mathrm{s1}≔\mathrm{Sum}\left(\frac{1}{n^2}\&comma;n=1..\mathrm{\infty }\right)$

$\mathrm{s1}≔\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$

$\mathrm{Converges}\left(\mathrm{s1}\right)$

$\mathrm{true}$

$\mathrm{s2}≔\mathrm{Sum}\left(\frac{{\left(-1\right)}^n}{n}\&comma;n=1..\mathrm{\infty }\right)$

$\mathrm{s2}≔\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^n}{n}$

$\mathrm{Converges}\left(\mathrm{s2}\right)$

$\mathrm{FAIL}$

$\mathrm{s3}≔\mathrm{Sum}\left(2^{-n}\&comma;n=1..\mathrm{\infty }\right)$

$\mathrm{s3}≔\sum _{n=1}^{\mathrm{\infty }}{2}^{-n}$

$\mathrm{Converges}\left(\mathrm{s3}\right)$

$\mathrm{true}$

$\mathrm{s4}≔\mathrm{convert}\left(\exp \left(x\right)\&comma;'\mathrm{FormalPowerSeries}'\right)$

$\mathrm{s4}≔\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$

$\mathrm{Converges}\left(\mathrm{s4}\right)$

$\mathrm{true}$

$\mathrm{s5}≔\mathrm{Sum}\left(\frac{x^n}{n}\&comma;n=1..\mathrm{\infty }\right)$

$\mathrm{s5}≔\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n}$

$\mathrm{Converges}\left(\mathrm{s5}\right)$

$\mathrm{FAIL}$

$\mathrm{ConvergenceRadius}\left(\mathrm{s5}\&comma;x\right)$

$\left|x\right|<1$

$\mathrm{ConvergenceRadius}\left(\frac{x^n}{n}\&comma;n\&comma;1\&comma;\mathrm{\infty }\&comma;x\right)$

$\left|x\right|<1$

$\mathrm{s6}≔\mathrm{Sum}\left(\mathrm{binomial}\left(2n\&comma;n\right)\&comma;n=0..\mathrm{\infty }\right)$

$\mathrm{s6}≔\sum _{n=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$

$\mathrm{ConvergenceRadius}\left(\mathrm{s6}\&comma;x\right)$

$\left|x\right|<0$

$\mathrm{s7}≔\mathrm{Sum}\left(\mathrm{binomial}\left(2n\&comma;n\right)x^n\&comma;n=0..\mathrm{\infty }\right)$

$\mathrm{s7}≔\sum _{n=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)x^n$

$\mathrm{ConvergenceRadius}\left(\mathrm{s7}\&comma;x\right)$

$\left|x\right|<\frac{1}{4}$

$\mathrm{s8}≔\mathrm{Sum}\left({\left(x-1\right)}^n{2}^{-n}\&comma;n=0..\mathrm{\infty }\right)$

$\mathrm{s8}≔\sum _{n=0}^{\mathrm{\infty }}{\left(x-1\right)}^n{2}^{-n}$

$\mathrm{ConvergenceRadius}\left(\mathrm{s8}\&comma;x\right)$

$\left|x-1\right|<2$

$\mathrm{s9}≔\mathrm{Sum}\left({\left(x^2-2\right)}^n\&comma;n=0..\mathrm{\infty }\right)$

$\mathrm{s9}≔\sum _{n=0}^{\mathrm{\infty }}{\left(x^2-2\right)}^n$

$\mathrm{ConvergenceRadius}\left(\mathrm{s9}\&comma;x\right)$

$\left|x^2-2\right|<1$

The SumTools[DefiniteSum][Converges] and SumTools[DefiniteSum][ConvergenceRadius] commands were introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.