title: “Taylor series”
date: 2024-06-04 18:22:26
updated: 2024-06-05 08:15:15
tag: [“notion”, “Math”]
categories: “Math”
mathjax: true
comments: true
description: ‘
$f(x) = f(x)+f’(x)x+\frac{f’’(x)}{2!}x^2+\frac{f’’’(x)}{3!}x^3+\dots++\frac{f^{(n)}(x)}{n!}x^n = \sum \limits_{k=0}^{n}\frac{f^{(k)}(x)}{k!}x^k$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $sin(x)$ | $0$ | $1$ |
| $f’(x)$ | $cos(x)$ | $1$ | $1$ |
| $f’’(x)$ | $-sin(x)$ | $0$ | $\frac{1}{2}$ |
| $f’’’(x)$ | $-cos(x)$ | $-1$ | $\frac{1}{3!}$ |
| $f^{n}(x)$ | $f^{n \mod 4}(x)$ | $f^{n \mod 4}(0)$ | $\frac{1}{n!}$ |
$sin(x) = x- \frac{x^3}{3!}+ \frac{x^5}{5!}- \frac{x^7}{7!}+ \frac{x^9}{9!}-\dots+\frac{x^{2n+1}}{(2n+1)!}(-1)^n = \sum \limits_{k=0}^{n}\frac{x^{2k+1}}{(2k+1)!}(-1)^k$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $cos(x)$ | $1$ | $1$ |
| $f’(x)$ | $-sin(x)$ | $0$ | $1$ |
| $f’’(x)$ | $-cos(x)$ | $-1$ | $\frac{1}{2}$ |
| $f’’’(x)$ | $sin(x)$ | $0$ | $\frac{1}{3!}$ |
| $f^n(x)$ | $f^{n \mod 4}(x)$ | $f^{n \mod 4}(0)$ | $\frac{1}{n!}$ |
$cos(x) = 1- \frac{x^2}{2!}+ \frac{x^4}{4!}- \frac{x^6}{6!}+ \frac{x^8}{8!}-\dots+\frac{x^{2n}}{(2n)!}(-1)^n = \sum \limits_{k=0}^{n}\frac{x^{2k}}{(2k)!}(-1)^k$
$tan(x) = x + \frac{x^3}{3} + \frac{2\cdot x^5}{15} + o(x^5)$
$\arcsin(x) = x+\frac{1\cdot x^3}{2\cdot 3}+\frac{1\cdot 3 \cdot x^5}{2\cdot 4 \cdot5} +\frac{1\cdot 3 \cdot 5 \cdot x^7}{2\cdot 4 \cdot 6 \cdot 7} + o(x^7)$
$\arccos(x) = \frac{\pi}{2}-\arcsin(x)=\frac{\pi}{2}-(x+\frac{1\cdot x^3}{2\cdot 3}+\frac{1\cdot 3 \cdot x^5}{2\cdot 4 \cdot5} +\frac{1\cdot 3 \cdot 5 \cdot x^7}{2\cdot 4 \cdot 6 \cdot 7} + o(x^7))$
$\arctan(x) = x- \frac{x^3}{3}+ \frac{x^5}{5}- \frac{x^7}{7}+ \frac{x^9}{9}-\dots+\frac{x^{2n+1}}{2n+1}(-1)^n = \sum \limits_{k=0}^{n}\frac{x^{2k+1}}{2k+1}(-1)^k$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $a^{x}$ | $1$ | $1$ |
| $f’(x)$ | $a^{x}\ln a$ | $\ln a$ | $\ln a$ |
| $f’’(x)$ | $a^{x}\ln^2 a$ | $\ln^2 a$ | $\frac{\ln^2a}{2}$ |
| $f’’’(x)$ | $a^{x}\ln^3 a$ | $\ln^3 a$ | $\frac{\ln^3a}{3!}$ |
| $f^n(x)$ | $a^{x}\ln^n a$ | $\ln^n a$ | $\frac{\ln^na}{n!}$ |
$a^x = 1+ x\ln a + \frac{(x\ln a)^2}{2!} + \frac{(x\ln a)^3}{3!} + \frac{(x\ln a)^4}{4!}+ \frac{(x\ln a)^5}{5!}+\dots+\frac{(x\ln a)^{n}}{n!} = \sum\limits_{k=0}^{n}\frac{(x\ln a)^k}{k!}$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $e^{x}$ | $1$ | $1$ |
| $f’(x)$ | $e^{x}$ | $1$ | $1$ |
| $f’’(x)$ | $e^{x}$ | $1$ | $\frac{1}{2}$ |
| $f’’’(x)$ | $e^{x}$ | $1$ | $\frac{1}{3!}$ |
| $f^n(x)$ | $e^{x}$ | $1$ | $\frac{1}{n!}$ |
$e^x = 1+ x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+ \frac{x^5}{5!}+\dots+\frac{x^{n}}{n!} = \sum\limits_{k=0}^{n}\frac{x^k}{k!}$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $e^{xi}$ | $1$ | $1$ |
| $f’(x)$ | $ie^{xi}$ | $i$ | $i$ |
| $f’’(x)$ | $i^2e^{xi}$ | $i^2$ | $\frac{i^2}{2}$ |
| $f’’’(x)$ | $i^3e^{xi}$ | $i^3$ | $\frac{i^3}{3!}$ |
| $f^n(x)$ | $i^ne^{xi}$ | $i^n$ | $\frac{i^n}{n!}$ |
$e^{xi} = 1+ xi + \frac{i^2x^2}{2!} + \frac{i^3x^3}{3!} + \frac{i^4x^4}{4!}+ \frac{i^5x^5}{5!}+\dots+\frac{i^nx^{n}}{n!} = \sum\limits_{k=0}^{n}\frac{i^kx^k}{k!}$
$e^{xi} = 1+ xi - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!}+ \frac{ix^5}{5!}+\dots+\frac{i^nx^{n}}{n!} = \sum\limits_{k=0}^{n}(\frac{x^{2k}}{(2k)!}+\frac{x^{2k+1}}{(2k+1)!}i)(-1)^k$
$e^{xi} = \sum\limits_{k=0}^{n}(\frac{x^{2k}}{(2k)!}+\frac{x^{2k+1}}{(2k+1)!}i)(-1)^k=\sum\limits_{k=0}^{n}\frac{x^{2k}}{(2k)!}(-1)^k+i\sum\limits_{k=0}^{n}\frac{x^{2k+1}}{(2k+1)!}(-1)^k$
$e^{xi}= cos(x)+sin(x)i$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $(1+x)^a$ | $1$ | $1$ |
| $f’(x)$ | $a\cdot(1+x)^{a-1}$ | $a$ | $a$ |
| $f’’(x)$ | $a\cdot (a-1) \cdot(1+x)^{a-2}$ | $a\cdot(a-1)$ | $\frac{a\cdot(a-1)}{2}$ |
| $f’’’(x)$ | $a\cdot (a-1) \cdot (a-2) \cdot(1+x)^{a-3}$ | $a\cdot (a-1) \cdot (a-2)$ | $\frac{a\cdot (a-1) \cdot (a-2)}{3!}$ |
| $f^n(x)$ | $A_a^n(1+x)^{a-n}$ | $A_a^n$ | $\frac{A_a^n}{n!}$ |
$(1+x)^a = 1+ax+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)}{3!}x^3+\dots+\frac{a(a-1)\cdots(a-n+1)}{n!}x^n$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $(1-x)^a$ | $1$ | $1$ |
| $f’(x)$ | $-a\cdot(1-x)^{a-1}$ | $a$ | $a$ |
| $f’’(x)$ | $a\cdot (a-1) \cdot(1-x)^{a-2}$ | $a\cdot(a-1)$ | $\frac{a\cdot(a-1)}{2}$ |
| $f’’’(x)$ | $-a\cdot (a-1) \cdot (a-2) \cdot(1-x)^{a-3}$ | $a\cdot (a-1) \cdot (a-2)$ | $\frac{a\cdot (a-1) \cdot (a-2)}{3!}$ |
| $f^n(x)$ | $(-1)^nA_a^n(1-x)^{a-n}$ | $(-1)^nA_a^n$ | $(-1)^n\frac{A_a^n}{n!}$ |
$(1-x)^a = 1-ax+\frac{a(a-1)}{2!}x^2-\frac{a(a-1)(a-2)}{3!}x^3+\dots+(-1)^n\frac{a(a-1)\cdots(a-n+1)}{n!}x^n$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $\frac{1}{1-x}$ | $1$ | $1$ |
| $f’(x)$ | $\frac{1}{1-x}^2$ | $1$ | $1$ |
| $f’’(x)$ | $2\frac{1}{1-x}^3$ | $2!$ | $1$ |
| $f’’’(x)$ | $3!\frac{1}{1-x}^4$ | $3!$ | $1$ |
| $f^n(x)$ | $n!\frac{1}{1-x}^{(n+1)}$ | $n!$ | $1$ |
$\frac{1}{1-x} = 1 + x + x^2 + x^3+\dots+x^n=\sum\limits_{i=0}^{n}x^i$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $\frac{1}{1+x}$ | $1$ | $1$ |
| $f’(x)$ | $-\frac{1}{1+x}^2$ | $-1$ | $-1$ |
| $f’’(x)$ | $2\frac{1}{1+x}^3$ | $2!$ | $1$ |
| $f’’’(x)$ | $-3!\frac{1}{1+x}^4$ | $-3!$ | $-1$ |
| $f^n(x)$ | $(-1)^nn!\frac{1}{1+x}^{(n+1)}$ | $(-1)^nn!$ | $(-1)^n$ |
$\frac{1}{1+x} = 1 - x + x^2 - x^3+\dots+(-1)^nx^n=\sum\limits_{i=0}^{n}x^i(-1)^n$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $\ln(1-x)$ | $0$ | $0$ |
| $f’(x)$ | $-\frac{1}{1-x}$ | $-1$ | $-1$ |
| $f’’(x)$ | $-\frac{1}{1-x}^2$ | $-1$ | $-\frac{1}{2}$ |
| $f’’’(x)$ | $-2\frac{1}{1-x}^3$ | $-2!$ | $-\frac{2!}{3!}=\frac{1}{3}$ |
| $f^4(x)$ | $-3!\frac{1}{1-x}^4$ | $-3!$ | $-\frac{1}{4}$ |
| $f^n(x)$ | $-(n-1)!\frac{1}{1-x}^{n}$ | $-(n-1)!$ | $-\frac{1}{n}$ |
$\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3}-\dots-\frac{x^n}{n}=-\sum\limits_{i=1}^{n}\frac{x^i}{n}$
| $f^n(x)$ | $f^n(0)$ | $a_n=\frac{f^n(0)}{n!}$ | |
|---|---|---|---|
| $f(x)$ | $\ln(1+x)$ | $0$ | $0$ |
| $f’(x)$ | $\frac{1}{1+x}$ | $1$ | $1$ |
| $f’’(x)$ | $-\frac{1}{1+x}^2$ | $-1$ | $-\frac{1}{2}$ |
| $f’’’(x)$ | $2\frac{1}{1+x}^3$ | $2!$ | $\frac{2!}{3!}=\frac{1}{3}$ |
| $f^4(x)$ | $-3!\frac{1}{1+x}^4$ | $-3!$ | $-\frac{1}{4}$ |
| $f^n(x)$ | $(-1)^{n+1}(n-1)!\frac{1}{1+x}^{n}$ | $(-1)^{n+1}(n-1)!$ | $(-1)^{n+1}\frac{1}{n}$ |
$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}-\dots+(-1)^{n+1}\frac{x^n}{n}=\sum\limits_{i=1}^{n}\frac{x^i}{n}(-1)^{i+1}$