Radius of Convergence (3)

Calculus Level 2

For the following series, determine the radius of convergence :

∑ n = 0 ∞ x n 2 \sum_{n=0}^{\infty} x^{n^2}


The answer is 1.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Chew-Seong Cheong
Dec 19, 2019

Applying ratio test lim ⁡ n → ∞ x ( n + 1 ) 2 x n 2 = lim ⁡ n → ∞ x 2 n + 1 < 1 ⟹ ∑ n = 0 ∞ x n 2 \displaystyle \lim_{n \to \infty} \frac {x^{(n+1)^2}}{x^{n^2}} = \lim_{n \to \infty} x^{2n+1} < 1 \implies \sum_{n=0}^\infty x^{n^2} converges for − 1 < x < 1 -1 < x < 1 . Therefore the radius of convergence is 1 \boxed 1 .

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