【微分積分】3-6-5 テイラー展開｜問題集
1.次の関数をマクローリン展開を求めなさい。
(1)\(\displaystyle \frac{1}{1+x}\)
\(\displaystyle =\frac{1}{1-(-x)}\)
\(\displaystyle =\sum_{n=0}^{\infty}(-x)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}(-1)^nx^n\)
\(\displaystyle =1-x+x^2-x^3+\cdots\)
\(\displaystyle =\sum_{n=0}^{\infty}(-x)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}(-1)^nx^n\)
\(\displaystyle =1-x+x^2-x^3+\cdots\)
(2)\(\displaystyle \frac{1}{3+2x}\)
\(\displaystyle =\frac{1}{3}・\frac{1}{1-\frac{-2x}{3}}\)
\(\displaystyle =\frac{1}{3}\sum_{n=0}^{\infty}\left(\frac{-2x}{3}\right)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{(-2)^n}{3^{n+1}}x^n\)
\(\displaystyle =\frac{1}{3}-\frac{2}{9}x+\frac{4}{27}x^2-\cdots\)
\(\displaystyle =\frac{1}{3}\sum_{n=0}^{\infty}\left(\frac{-2x}{3}\right)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{(-2)^n}{3^{n+1}}x^n\)
\(\displaystyle =\frac{1}{3}-\frac{2}{9}x+\frac{4}{27}x^2-\cdots\)
(3)\(\displaystyle \frac{1}{1-x^2}\)
\(\displaystyle =\sum_{n=0}^{\infty}(x^2)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}x^{2n}\)
\(\displaystyle =1+x^2+x^4+\cdots\)
\(\displaystyle =\sum_{n=0}^{\infty}x^{2n}\)
\(\displaystyle =1+x^2+x^4+\cdots\)
(4)\(\displaystyle \frac{x}{1+x^2}\)
\(\displaystyle =x・\frac{1}{1-(-x^2)}\)
\(\displaystyle =x\sum_{n=0}^{\infty}(-x^2)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}(-1)^nx^{2n+1}\)
\(\displaystyle =x-x^3+x^5-\cdots\)
\(\displaystyle =x\sum_{n=0}^{\infty}(-x^2)^n\)
\(\displaystyle =\sum_{n=0}^{\infty}(-1)^nx^{2n+1}\)
\(\displaystyle =x-x^3+x^5-\cdots\)
(5)\(\displaystyle \frac{x-1}{2x^2+3x-2}\)
\(\displaystyle =\frac{1}{5}\left(\frac{3}{x+2}-\frac{1}{2x-1}\right)\)
\(\displaystyle =\frac{1}{5}\left(\frac{3}{2}・\frac{1}{1-\frac{-x}{2}}+\frac{1}{1-2x}\right)\)
\(\displaystyle =\frac{1}{5}\left\{\frac{3}{2}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n+\sum_{n=0}^{\infty}(2x)^n\right\}\)
\(\displaystyle =\frac{1}{5}\left\{\frac{3(-1)^n}{2^{n+1}}+2^n\right\}x^n\)
\(\displaystyle =\frac{1}{2}+\frac{5}{20}x+\frac{35}{40}x^2+\cdots\)
\(\displaystyle =\frac{1}{5}\left(\frac{3}{2}・\frac{1}{1-\frac{-x}{2}}+\frac{1}{1-2x}\right)\)
\(\displaystyle =\frac{1}{5}\left\{\frac{3}{2}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n+\sum_{n=0}^{\infty}(2x)^n\right\}\)
\(\displaystyle =\frac{1}{5}\left\{\frac{3(-1)^n}{2^{n+1}}+2^n\right\}x^n\)
\(\displaystyle =\frac{1}{2}+\frac{5}{20}x+\frac{35}{40}x^2+\cdots\)
(6)\(\sin^2x\)
\(\displaystyle =\frac{1-\cos2x}{2}\)
\(\displaystyle =\frac{1}{2}-\frac{1}{2}\sum_{n=0}^{\infty}\frac{(-1)^n(2x)^{2n}}{(2n)!}\)
\(\displaystyle =\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n-1}}{(2n)!}x^{2n}\)
\(\displaystyle =x^2-\frac{1}{3}x^4+\frac{2}{45}x^6-\cdots\)
\(\displaystyle =\frac{1}{2}-\frac{1}{2}\sum_{n=0}^{\infty}\frac{(-1)^n(2x)^{2n}}{(2n)!}\)
\(\displaystyle =\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n-1}}{(2n)!}x^{2n}\)
\(\displaystyle =x^2-\frac{1}{3}x^4+\frac{2}{45}x^6-\cdots\)
(7)\(\sinh x\)
\(\displaystyle =\frac{e^x-e^{-x}}{2}\)
\(\displaystyle =\frac{1}{2}\left(\sum_{n=0}^{\infty}\frac{x^n}{n!}-\sum_{n=0}^{\infty}\frac{(-x)^n}{n!}\right)\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{1}{(2n+1)!}x^{2n+1}\)
\(\displaystyle =x+\frac{x^3}{6}+\frac{x^5}{120}+\cdots\)
\(\displaystyle =\frac{1}{2}\left(\sum_{n=0}^{\infty}\frac{x^n}{n!}-\sum_{n=0}^{\infty}\frac{(-x)^n}{n!}\right)\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{1}{(2n+1)!}x^{2n+1}\)
\(\displaystyle =x+\frac{x^3}{6}+\frac{x^5}{120}+\cdots\)
(8)\(\log(1+x-6x^2)\)
\(\displaystyle =\log(1+3x)+\log(1-2x)\)
\(\displaystyle =\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(3x)^n}{n}+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(-2x)^n}{n}\)
\(\displaystyle =\sum_{n=1}^{\infty}(-1)^{n-1}\frac{3^n+(-2)^n}{n}x^n\)
\(\displaystyle =x-\frac{13}{2}x^2+\frac{19}{3}x^3-\cdots\)
\(\displaystyle =\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(3x)^n}{n}+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(-2x)^n}{n}\)
\(\displaystyle =\sum_{n=1}^{\infty}(-1)^{n-1}\frac{3^n+(-2)^n}{n}x^n\)
\(\displaystyle =x-\frac{13}{2}x^2+\frac{19}{3}x^3-\cdots\)
(9)\(x^2e^{2x}\)
\(\displaystyle =x^2\sum_{n=0}^{\infty}\frac{(2x)^n}{n!}\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{2^n}{n!}x^{n+2}\)
\(\displaystyle =x^2+2x^3+2x^4+\frac{4}{3}x^5+\cdots\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{2^n}{n!}x^{n+2}\)
\(\displaystyle =x^2+2x^3+2x^4+\frac{4}{3}x^5+\cdots\)
(10)\(2^x\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{(x\log2)^n}{n!}\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{(\log2)^n}{n!}x^n\)
\(\displaystyle =1+(\log2)x+\frac{(\log2)^2}{2}x^2+\frac{(\log2)^3}{6}x^3+\cdots\)
\(\displaystyle =\sum_{n=0}^{\infty}\frac{(\log2)^n}{n!}x^n\)
\(\displaystyle =1+(\log2)x+\frac{(\log2)^2}{2}x^2+\frac{(\log2)^3}{6}x^3+\cdots\)
次の学習に進もう！