数列(sequences)

確認問題

1.

次の数列の極限を求めよう．

(a) $${\{a_{n}\} = \{\sqrt{n}\}}$$ (b) $${\{a_{n}\} = \{\frac{n+1}{n^{2}}\}}$$ (c) $${\{a_{n}\} = \{\frac{n^{2}}{n + 1}\}}$$

(d) $${\{a_{n}\} = \{\frac{2^{n} - 1}{2^{n}}\}}$$ (e) $${\{a_{n}\} = \{\frac{1}{n} - \frac{1}{n+1}\}}$$

2.

次の数列は有界か調べよう．また，単調性についても調べよう．

(a) $${\{a_{n}\} = \{\frac{2}{n}\}}$$ (b) $${\{a_{n}\} = \{\sqrt{4 - \frac{1}{n}}\}}$$

3.

次の漸化式で定義される数列 $\{a_{n}\}$ の一般項を求めよう．

(a) $${a_{1} = 1, a_{n+1} = \frac{1}{n+1}a_{n}, n \geq 1}$$ (b) $${a_{1} = 1, a_{n+1} = a_{n} + 2, n \geq 1}$$

(c) $${a_{1} = 1, a_{n+1} = a_{n} + 2n + 1, n \geq 1}$$

演習問題

1.

次の数列の極限を求めよう．

(a) $${\{a_{n}\} = \{n^4 - 3n^3\}}$$ (b) $${\{a_{n}\} = \{\frac{3n^{2}+5}{4n^{3} - 1}\}}$$ (c) $${\{a_{n}\} = \{\frac{1 - n}{n - \sqrt{n}}\}}$$

(d) $${\{a_{n}\} = \{\frac{n(n+2)}{n+1} - \frac{n^{3}}{n^{2}+1}\}}$$ (e) $${\{a_{n}\} = \{\sqrt{n+1} - \sqrt{n}\}}$$

2.

$a > 0$ のとき $${\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1}$$ を証明しよう．

3.

$${\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1}$$を用いて次の極限値を求めよう．

(a) $a > b > 0$のとき， $${\{ (a^n + b^n)^{\frac{1}{n}}\}}$$ (b) $${\{a_{n}\} = \{(1+2^{n}+3^{n})^{\frac{1}{n}}\}}$$