>>650

の解答を自分なりに補ってみました：


以下のべき級数の収束半径を求める。

Σ_{n=0}^{∞} a_n * x^n = x - x^2/2 + x^4/4 - x^5/5 + x^7/7 - x^8/8 ± …

(|a_1|)^(1/1) = 1^(1/1) = 1
(|a_2|)^(1/2) = (1/2)^(1/2) = 1/sqrt(2)
(|a_3|)^(1/3) = 0^(1/3) = 0
(|a_4|)^(1/4) = (1/4)^(1/4) = 1/sqrt(2)
(|a_5|)^(1/5) = (1/5)^(1/5)
(|a_6|)^(1/6) = 0^(1/6) = 0
(|a_7|)^(1/7) = (1/7)^(1/7)
(|a_8|)^(1/8) = (1/8)^(1/8)
…

n ≡ 0 (mod 3) でないとき、

(|a_n|)^(1/n) = (1/n)^(1/n) = 1/(n)^(1/n)

n ≡ 0 (mod 3) であるとき、

(|a_n|)^(1/n) = 0

n ≡ 0 (mod 3) でないとき、

(|a_n|)^(1/n) = 1/(n)^(1/n) ≦ 1

n ≡ 0 (mod 3) であるとき、

(|a_n|)^(1/n) = 0 ≦ 1

よって、 (|a_n|)^(1/n) ≦ 1 for all n ∈ {1, 2, 3, …}