第2章 2.1

(1)

$\lim_{n \to \infty} \frac{2n}{1-3n} = \lim_{n \to \infty} \frac{2}{\dfrac{1}{n}-3} = -\frac{2}{3} \quad \text{収束}$

(2)

$\lim_{n \to \infty} \frac{3n^2+1}{2n^2-3} = \lim_{n \to \infty} \frac{3+\dfrac{1}{n^2}}{2-\dfrac{3}{n^2}} = \frac{3}{2} \quad \text{収束}$

(3)

$\lim_{n \to \infty} \frac{n(n+4)}{n(n+1)(n+3)} = \lim_{n \to \infty} \frac{\dfrac{1}{n}\left(1+\dfrac{4}{n}\right)}{\left(1+\dfrac{1}{n}\right)\left(1+\dfrac{3}{n}\right)} = 0 \quad \text{収束}$

(4)

$\lim_{n \to \infty} \frac{1+n^2}{1+n} = \lim_{n \to \infty} \frac{\dfrac{1}{n^2}+1}{\dfrac{1}{n^2}+\dfrac{1}{n}} = \infty \quad \text{正の無限大に発散}$

(5)

$\lim_{n \to \infty} \frac{(-1)^n 2n}{n-3} = \lim_{n \to \infty} \frac{(-1)^n 2}{1-\dfrac{3}{n}} = (-1)^n 2 \quad \text{n \text{により振動}}$

(6)

$\lim_{n \to \infty} \frac{(-2)^n}{3^n-1} = \lim_{n \to \infty} \frac{\left(\dfrac{-2}{3}\right)^n}{1-\dfrac{1}{3^n}} = 0 \quad \text{収束}$

(7)

$\lim_{n \to \infty} \sqrt{n}(\sqrt{n+1} - \sqrt{n}) = \lim_{n \to \infty} \frac{\sqrt{n}(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} = \lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n+1} + \sqrt{n}} = \lim_{n \to \infty} \frac{1}{\sqrt{1 + \dfrac{1}{n}} + 1} = \frac{1}{2}$

収束

(8)

$\lim_{n \to \infty} \frac{1}{n} \cos n\pi = 0$

収束