lectures:maclaurin_exp
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| lectures:maclaurin_exp [2021/02/17 12:38] – [主な関数のMaclaurin展開] kimi | lectures:maclaurin_exp [2022/08/23 13:34] (現在) – 外部編集 127.0.0.1 | ||
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| ===== 主な関数のMaclaurin展開 ===== | ===== 主な関数のMaclaurin展開 ===== | ||
| - | * $\sin x$\\ $$\sin x=x-\frac{1}{6}x^3+\frac{1}{120}x^5-\cdots+\frac{(-1)^m}{(2m+1)!}x^{2m+1}+\cdots$$ | + | ==== $\sin x$ ==== |
| - | | + | $$ |
| + | \sin x=x-\frac{1}{6}x^3+\frac{1}{120}x^5-\cdots+\frac{(-1)^m}{(2m+1)!}x^{2m+1}+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-\infty<x< | ||
| + | ==== $\cos x$ ==== | ||
| + | $$ | ||
| + | \cos x=1-\frac{1}{2}x^2+\frac{1}{24}x^4-\frac{1}{720}x^6+\cdots+\frac{(-1)^m}{(2m)!}x^{2m}+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-\infty< | ||
| + | ==== $\exp x$ ==== | ||
| + | $$ | ||
| + | e^x=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+\frac{1}{24}x^4+\frac{1}{120}x^5+\frac{1}{720}x^6+\cdots+\frac{1}{n!}x^n+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-\infty< | ||
| + | ==== $\ln(1+x)$ ==== | ||
| + | $$ | ||
| + | \ln(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5+\cdots+\frac{(-1)^{n-1}}{n}x^n+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-1< | ||
| + | ==== $\displaystyle\frac{1}{1-x}$ ==== | ||
| + | $$ | ||
| + | \frac{1}{1-x}=1+x+x^2+x^3+x^4+x^5+\cdots+x^n+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-1< | ||
| + | ==== $(1+x)^\alpha$ ==== | ||
| + | $$ | ||
| + | (1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha -1)}{2}x^2+\frac{\alpha(\alpha -1)(\alpha -2)}{6}x^3+\cdots | ||
| + | $$ | ||
| + | $$ | ||
| + | \cdots+\frac{\alpha(\alpha -1)(\alpha -2)\cdots(\alpha -n+1)}{n!}x^n+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-1< | ||
| + | ==== $\displaystyle\frac{1}{\sqrt{1-x}}$ ==== | ||
| + | $$ | ||
| + | \frac{1}{\sqrt{1-x}}=1+\frac{1}{2}x+\frac{3}{8}x^2+\frac{5}{16}x^3+\cdots+\frac{(2n-1)!!}{2^nn!}x^n+\cdots | ||
| + | $$ | ||
| + | 収束半径は$(-1< | ||
lectures/maclaurin_exp.1613533134.txt.gz · 最終更新: 2022/08/23 13:34 (外部編集)