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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.

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).

Find R for Σ xⁿ/n!.
  1. |a_{n+1}/aₙ| = |x|/(n+1) → 0 for all x

Answer: R = ∞ (converges for all x).

Related formulas

Power Series

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