第3章 3.2

(1)

$x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots + \frac{x^{2n+1}}{2n+1} + \cdots$

$\sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1}$

$r = \lim_{n \to \infty} \left| \dfrac{\dfrac{1}{2n+1}}{\dfrac{1}{2n+3}} \right| = \lim_{n \to \infty} \left| \frac{2n+3}{2n+1} \right| = 1$

(2)

$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots + \frac{x^{2n}}{(2n)!}$

$\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$

$r = \lim_{n \to \infty} \left| \dfrac{\dfrac{1}{(2n)!}}{\dfrac{1}{\{2(n+1)\}!}} \right| = \lim_{n \to \infty} \left| \frac{\{2(n+1)\}!}{(2n)!} \right| = \lim_{n \to \infty} |(2n+1)(2n+2)| = \infty$

(3)

$x + \frac{2!}{3}x^2 + \frac{3!}{5}x^3 + \cdots + \frac{n!}{2n-1}x^n + \cdots$

$\sum_{n=1}^{\infty} \frac{n!}{2n-1}x^n$

$r = \lim_{n \to \infty} \left| \dfrac{\dfrac{n!}{2n-1}}{\dfrac{(n+1)!}{2n+1}} \right| = \lim_{n \to \infty} \left| \frac{(2n+1)n!}{(2n-1)(n+1)!} \right| = \lim_{n \to \infty} \left| \frac{2n+1}{(2n-1)(n+1)} \right| = \lim_{n \to \infty} \left| \frac{2n+1}{2n^2+n-1} \right| = 0$