【微分積分】1-1-6 増加速度の比較｜問題集

\(\displaystyle =\lim_{n\to\infty}\frac{(\frac{2}{3})^n+\frac{n^{10}}{3^n}}{1+\frac{n^{10}}{3^n}}\)
\(\displaystyle \lim_{n\to\infty}\frac{n^{10}}{3^n}=0\)より、
\(\displaystyle =\frac{0+0}{1+0}\)
\(=0\)

\(\displaystyle =\lim_{n\to\infty}\frac{\frac{2^n}{n!}+1}{\frac{3^n}{n!}+1}\)
\(\displaystyle \lim_{n\to\infty}\frac{2^n}{n!}=\lim_{n\to\infty}\frac{3^n}{n!}=0\)より、
\(\displaystyle =\frac{0+1}{0+1}\)
\(=1\)

\(\displaystyle a_n=\frac{n!}{2^{n^2}}\)とおくと、
\(\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_n}\)
\(\displaystyle =\lim_{n\to\infty}\frac{(n+1)!}{2^{(n+1)^2}}・\frac{2^{n^2}}{n!}\)
\(\displaystyle =\lim_{n\to\infty}\frac{n+1}{2^{2n+1}}\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{2}・\frac{n+1}{4^n}\)
\(=0\)
\(\displaystyle \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=a\)より、
\(\displaystyle \lim_{n\to\infty}\frac{n!}{2^{n^2}}=0\)