1/(1-x)のマクローリン展開(ただし-1<x<1)

$\frac{1}{1-x}$のマクローリン展開($\mathrm{た}\mathrm{だ}\mathrm{し}-1<x<1$)

a点まわりのテイラー展開

$$\begin{array}{rcl}f(x)& =& \sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(a)}{x!}(x-a{)}^k\dots a\mathrm{点}\mathrm{ま}\mathrm{わ}\mathrm{り}\mathrm{の}\mathrm{テ}\mathrm{イ}\mathrm{ラ}\mathrm{ー}\mathrm{展}\mathrm{開}\\ & =& \frac{1}{0!}{f}^{(0)}(a)(x-a{)}^0+\frac{1}{1!}{f}^{(1)}(a)(x-a{)}^1+\frac{1}{2!}{f}^{(2)}(a)(x-a{)}^2+\cdots \\ & & \dots f^{(n)}(x):f(x)\mathrm{の}n\mathrm{階}\mathrm{微}\mathrm{分}\end{array}$$

マクローリン展開(0点まわりのテイラー展開)

$$\begin{array}{rcl}f(x)& =& {\sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(a)}{x!}(x-a{)}^k|}_{a=0}\\ & =& \sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(0)}{x!}(x-0{)}^k\\ & =& \sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(0)}{x!}(x{)}^k\\ & =& \frac{1}{0!}{f}^{(0)}(0)x^0+\frac{1}{1!}{f}^{(1)}(0)x+\frac{1}{2!}{f}^{(2)}(0)x^2+\cdots \end{array}$$

マクローリン展開(0点まわりのテイラー展開)

$$\begin{array}{rcl}\frac{1}{1-x}& =& \sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}{f}^{\left(k\right)}\left(0\right)x^k\dots (\mathrm{た}\mathrm{だ}\mathrm{し}-1<x<1)\\ & =& \frac{1}{0!}{\left[{(1-x)}^{-1}\right]}_{x=0}{x}^0+\frac{1}{1!}{[-{(1-x)}^{-1-1}(-1)]}_{x=0}{x}^1+\frac{1}{2!}{[-2{(1-x)}^{-2-1}(-1)]}_{x=0}{x}^2+\frac{1}{3!}{[-6{(1-x)}^{-3-1}(-1)]}_{x=0}{x}^3+\cdots \\ & =& 1+x+x^2+x^3+\cdots \end{array}$$