2-1-2 無限等比数列(要点)

【等比数列\(\{r^n\}\)の極限】

(1)\(r>1\)のとき、\(\displaystyle \lim_{n\to\infty}r_n=\infty\)
(2)\(r=1\)のとき、\(\displaystyle \lim_{n\to\infty}r_n=1\)
(3)\(|r|<1\)のとき、\(\displaystyle \lim_{n\to\infty}r_n=0\)
(4)\(r\leqq -1\)のとき、振動する。

【例題】次の極限を求めなさい。

(1)\(\displaystyle \lim_{n\to\infty}\left(\frac{\sqrt{3}}{2}\right)^n\)

\(=0\)

(2)\(\displaystyle \lim_{n\to\infty}\left(\frac{\sqrt{5}}{2}\right)^n\)

\(=\infty\)

(3)\(\displaystyle \lim_{n\to\infty}\frac{2^{n+1}}{2^n+1}\)

\(\displaystyle =\lim_{n\to\infty}\frac{2}{1+(\frac{1}{2})^n}\)
\(=2\)

(4)\(\displaystyle \lim_{n\to\infty}\frac{5^n}{3^n+4^n}\)

\(\displaystyle =\lim_{n\to\infty}\frac{(\frac{5}{4})^n}{(\frac{3}{4})^n+1}\)
\(=\infty\)

【例題】数列\(\displaystyle \left\{\frac{2r^n}{1+r^{n+1}}\right\}\)の極限を次の場合について求めなさい。

(1)\(r>1\)

\(\displaystyle \lim_{n\to\infty}\frac{2r^n}{1+r^{n+1}}\)
\(\displaystyle =\lim_{n\to\infty}\frac{2}{(\frac{1}{r})^n+r}\)
\(\displaystyle =\frac{2}{r}\)

(2)\(r=1\)

\(\displaystyle \lim_{n\to\infty}\frac{2r^n}{1+r^{n+1}}\)
\(\displaystyle =\frac{2}{1+1}\)
\(\displaystyle =1\)

(3)\(|r|<1\)

\(\displaystyle \lim_{n\to\infty}\frac{2r^n}{1+r^{n+1}}\)
\(\displaystyle =\frac{0}{1+0}\)
\(\displaystyle =0\)

(4)\(r<-1\)

\(\displaystyle \lim_{n\to\infty}\frac{2r^n}{1+r^{n+1}}\)
\(\displaystyle =\lim_{n\to\infty}\frac{2}{(\frac{1}{r})^n+r}\)
\(\displaystyle =\frac{2}{r}\)