【微分積分】1-3-2 単調数列の収束性｜問題集
1.次の数列の極限を求めなさい。
(1)\(\displaystyle \lim_{n\to\infty}\left(\frac{n+2}{n+1}\right)^n\)
\(\displaystyle =\lim_{n\to\infty}\left(1+\frac{1}{n+1}\right)^n\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1+\frac{1}{n+1}\right)^{n+1}\right\}^{\frac{n}{n+1}}\)
\(m=n+1\)とおくと、\(n\to\infty\)のとき\(m\to\infty\)なので、
\(\displaystyle =\lim_{m\to\infty}\left\{\left(1+\frac{1}{m}\right)^m\right\}^{\frac{m-1}{m}}\)
\(=e^1\)
\(=e\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1+\frac{1}{n+1}\right)^{n+1}\right\}^{\frac{n}{n+1}}\)
\(m=n+1\)とおくと、\(n\to\infty\)のとき\(m\to\infty\)なので、
\(\displaystyle =\lim_{m\to\infty}\left\{\left(1+\frac{1}{m}\right)^m\right\}^{\frac{m-1}{m}}\)
\(=e^1\)
\(=e\)
(2)\(\displaystyle \lim_{n\to\infty}\left(1-\frac{1}{n+1}\right)^n\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1-\frac{1}{n+1}\right)^{n+1}\right\}^{\frac{n}{n+1}}\)
\(m=n+1\)とおくと、\(n\to\infty\)のとき\(m\to\infty\)なので、
\(\displaystyle =\lim_{m\to\infty}\left\{\left(1-\frac{1}{m}\right)^m\right\}^{\frac{m-1}{m}}\)
\(\displaystyle =\left(\frac{1}{e}\right)^1\)
\(\displaystyle =\frac{1}{e}\)
\(m=n+1\)とおくと、\(n\to\infty\)のとき\(m\to\infty\)なので、
\(\displaystyle =\lim_{m\to\infty}\left\{\left(1-\frac{1}{m}\right)^m\right\}^{\frac{m-1}{m}}\)
\(\displaystyle =\left(\frac{1}{e}\right)^1\)
\(\displaystyle =\frac{1}{e}\)
(3)\(\displaystyle \lim_{n\to\infty}\left(1-\frac{1}{n^2}\right)^n\)
\(\displaystyle =\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\left(1-\frac{1}{n}\right)^n\)
\(\displaystyle =e・\frac{1}{e}\)
\(=1\)
\(\displaystyle =e・\frac{1}{e}\)
\(=1\)
次の学習に進もう！