Radius of Convergence Calculator

? Understanding Radius of Convergence

Definition

For a power series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{a}_n(x-c{)}^n}$, the radius of convergence $R$ is a non‑negative real number or $\mathrm{\infty }$ such that the series converges absolutely for $|x-c|<R$ and diverges for $|x-c|>R$. At the endpoints $|x-c|=R$, convergence must be checked separately.

How to find $R$ – Ratio & Root Tests

Both tests compute a limit $L$ (if it exists) and then $R=1/L$ (with the conventions $1/0=\mathrm{\infty }$ and $1/\mathrm{\infty }=0$).

• Ratio test: ${\displaystyle L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|}$
• Root test: ${\displaystyle L=\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{|a_n|}}$

If the limit does not exist, one may use the superior limit (limsup) for the root test, but our calculator assumes the limit exists or can be approximated numerically.

Examples of common series

$a_n$	$L$ (ratio test)	$R$	Interval (center $c=0$)
$1/n!$	$0$	$\mathrm{\infty }$	$(-\mathrm{\infty },\mathrm{\infty })$
$1/n^p$ (any fixed $p$)	$1$	$1$	$(-1,1)$
$1$ (geometric)	$1$	$1$	$(-1,1)$
$n!$	$\mathrm{\infty }$	$0$	only $x=0$
$\frac{n!}{n^n}$	$1/e$	$e$	$(-e,e)$
$\frac{1}{2^n}$	$1/2$	$2$	$(-2,2)$

What about endpoints?

The radius gives the open interval of absolute convergence. At $x=c\pm R$, the series may converge conditionally or diverge. For example, $\sum x^n/n$ has $R=1$, converges at $x=-1$ (alternating harmonic) but diverges at $x=1$. Our calculator does not check endpoints automatically – you must apply additional tests (alternating series test, p‑test, etc.).

Numerical stability for huge terms (e.g. factorials)

When $|a_n|$ grows or decays extremely fast (like $n!$), direct evaluation causes overflow/underflow. Our numeric mode now uses logarithmic scaling:
${\displaystyle \ln \left|\frac{a_{n+1}}{a_n}\right|=\ln |a_{n+1}|-\ln |a_n|}$ and then exponentiates,
${\displaystyle \frac{1}{n}\ln |a_n|}$ for the root test. This avoids numerical overflow and gives reliable estimates even for $n$ up to millions.

When to use Auto / Numeric mode

• Auto: Tries symbolic limit first (nerdamer). If successful, exact $R$ is shown. If symbolic fails, it automatically falls back to the logarithmic numeric method.
• Numeric only: Skips symbolic, directly uses log‑scale numeric approximation. Useful for very complicated expressions where symbolic integration is hopeless.