数列(sequences)

演習問題

1.: 次の数列の極限を求めよう．
(a): $\displaystyle{\{a_{n}\} = \{n^4 - 3n^3\}}$
(b): $\displaystyle{\{a_{n}\} = \{\frac{3n^{2}+5}{4n^{3} - 1}\}}$
(c): $\displaystyle{\{a_{n}\} = \{\frac{1 - n}{n - \sqrt{n}}\}}$
(d): $\displaystyle{\{a_{n}\} = \{\frac{n(n+2)}{n+1} - \frac{n^{3}}{n^{2}+1}\}}$
(e): $\displaystyle{\{a_{n}\} = \{\sqrt{n+1} - \sqrt{n}\}}$
2.: のとき $\displaystyle{\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1}$ を証明しよう．
3.: $\displaystyle{\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1}$ を用いて次の極限値を求めよう．
(a): のとき， $\displaystyle{\lim_{n \rightarrow \infty} (a^n + b^n)^{\frac{1}{n}}}$
(b): $\displaystyle{\{a_{n}\} = \{(1+2^{n}+3^{n})^{\frac{1}{n}}\}}$