CalcRef

Sequences & Series · Power & Taylor Series

Radius of Convergence

R=lim⁡n→∞∣cncn+1∣orR=1lim sup⁡∣cn∣1/nR = \lim_{n \to \infty} \left|\frac{c_n}{c_{n+1}}\right| \quad \text{or} \quad R = \frac{1}{\limsup |c_n|^{1/n}}

The radius of convergence R determines where a power series converges absolutely. Often found via the ratio or root test.

  • 2 worked examples
  • 18 in Sequences & Series

Worked examples

Find R for Σ n! xⁿ.
  1. |a_{n+1}/aₙ| = (n+1)|x|
  2. lim (n+1)|x| = ∞ for any x ≠ 0

Answer: R = 0 (converges only at x = 0).

1 more worked examplePremium

Find R for Σ xⁿ/n!.

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

Related formulas

Power Series

See all 18 Sequences & Series formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Radius of Convergence?

The radius of convergence R determines where a power series converges absolutely. Often found via the ratio or root test. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Radius of Convergence?

Worked example. Find R for Σ n! xⁿ. |a_{n+1}/aₙ| = (n+1)|x|. lim (n+1)|x| = ∞ for any x ≠ 0. Answer: R = 0 (converges only at x = 0).

Practice

Quiz yourself on Sequences & Series

Radius of Convergence is one of 18 Sequences & Series formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 18 Sequences & Series formulas