Radius of convergence from coefficients calculator

It returns the estimated radius, every computed ratio, the tail values used for the estimate, and a stabilization check. That makes the result useful both as a quick calculation and as evidence you can inspect. The calculator is deterministic, runs without external data, and supports positive, negative, or alternating nonzero coefficients.

Turn a coefficient sequence into a convergence radius

For a power series written as the sum of a_n times (x minus c) to the nth power, the coefficient-ratio form of the ratio test gives the radius as R = lim |a_n / a_(n+1)|, provided that limit exists. Enter the coefficients in ascending power order: the first value is a_0, the second is a_1, and so on. The calculator forms an absolute ratio from each adjacent pair, so alternating signs do not change the estimate. It then averages the requested number of ratios at the end of the sequence. That tail average is reported as the radius estimate because later coefficients normally carry the most useful evidence about the limiting behavior. The center c is not needed: translating the variable changes where the interval of convergence sits, but it does not change its radius. Use enough coefficients to reveal a stable pattern. Three values technically produce ratios, but a longer sequence gives a much more meaningful view of whether the ratios approach a common limit.

Read the stabilization diagnostic before trusting the estimate

A finite list cannot prove an infinite limit, so the result includes more than one headline number. The ratios field shows every |a_n / a_(n+1)| value, while tail_ratios isolates the final window used in the estimate. The relative_spread compares the largest and smallest tail ratios with their average. When that spread is no greater than the selected tolerance, stabilized is true. This is a numerical diagnostic, not a theorem: a slowly changing sequence may look stable for several terms and drift later, while a genuinely convergent sequence may need many terms before its spread becomes small. Adjust tail_terms to balance recency against noise. A larger window asks more ratios to agree; a smaller window reacts more quickly to late behavior. Set tolerance according to the precision justified by the coefficients rather than forcing a favorable label. If stabilized is false, inspect the listed ratios, provide more coefficients, or use a different convergence theorem instead of presenting the tail average as an established exact radius.

Know when the coefficient-ratio method applies

This calculator deliberately requires finite, nonzero coefficients. A zero denominator makes a consecutive coefficient ratio undefined, and isolated or recurring zero coefficients can require the root test, a subsequence argument, or algebraic recognition of the series. Rejecting those inputs prevents an apparently ordinary number from hiding that mathematical issue. The method is especially effective for geometric-type coefficients, factorial expressions after numerical evaluation, and sequences whose adjacent ratios visibly settle toward a positive constant. If ratios grow without bound, the theoretical radius may be infinite; if they tend toward zero, the radius may be zero. A finite sample, however, cannot certify either infinite behavior, so this tool returns a finite tail estimate and exposes the ratios rather than guessing a symbolic infinity. After finding R, remember that the ratio test describes absolute convergence for |x-c| less than R and divergence for |x-c| greater than R. The endpoints x = c plus or minus R must be checked separately because the radius calculation alone says nothing decisive about them.