Interval, radius and centre of convergence of Σ2ⁿxⁿ/n!

Short answer

Ratio test: |u_{n+1}/u_n| = 2|x|/(n+1) → 0 for all x. Radius R = ∞, centre at 0, interval (-∞, ∞).

Find the interval of convergence, radius of convergence and centre of convergence of the power series Σ 2ⁿxⁿ/n!.

The series is Σ (n = 0 to ∞) 2ⁿ xⁿ/n!.

The series is in powers of x = (x − 0), so the centre of convergence is x = 0.

|uₙ₊₁/uₙ| = 2|x|/(n + 1) → 0   as n → ∞, for every fixed x

The limit 0 < 1 for every x, so the series converges for all x.

Radius of convergence R = ∞.

Interval of convergence: (−∞, ∞).

(In fact Σ (2x)ⁿ/n! = e^(2x), which is defined for every real x.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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