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Math formulas: Taylor and Maclaurin Series

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Definition of Taylor series:

f(x)=f(a)+f(a)(xa)+f(a)(xa)22!++f(n1)(a)(xa)n1(n1)!+Rn f(x) = f(a) + f'(a)(x-a) + \frac{f{'}{'}(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n
Rn=f(n)(ξ)(xa)nn! where aξx, ( Lagrangue’s form ) R_n = \frac{f^{(n)}(\xi)(x-a)^n}{n!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Lagrangue's form )}
Rn=f(n)(ξ)(xξ)n1(xa)(n1)! where aξx, ( Cauch’s form ) R_n = \frac{f^{(n)}(\xi)(x-\xi)^{n-1}(x-a)}{(n-1)!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Cauch's form )}

This result holds if f(x)f(x) has continuous derivatives of order nn at last. If limn+Rn=0\lim_{n \to +\infty}R_n = 0, the infinite series obtained is called Taylor series for f(x)f(x) about x=ax = a. If a=0a = 0 the series is often called a Maclaurin series.

Binomial series

(a+x)n=an+nan1+n(n1)2!an2x2+n(n1)(n2)3!an3x3+=an+(n1)an1x+(n2)an2x2+(n3)an3x3+ \begin{aligned} (a + x)^n &= a^n + na^{n-1} + \frac{n(n-1)}{2!} a^{n-2}x^2 + \frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+\cdots \\ &= a^n + { n \choose 1} a^{n-1}x + { n \choose 2} a^{n-2}x^2 + { n \choose 3} a^{n-3}x^3 + \cdots \end{aligned}

Special cases of binomial series

(1+x)1=1x+x2x3+1<x<1 (1 + x)^{-1} = 1 - x + x^2 -x^3 + \cdots \quad -1 < x < 1
(1+x)2=12x+3x24x3+1<x<1 (1 + x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \cdots \quad -1 < x < 1
(1+x)3=13x+6x210x3+1<x<1 (1 + x)^{-3} = 1 - 3x + 6x^2 - 10x^3 + \cdots \quad -1 < x < 1
(1+x)1/2=112x+1324x2135246x3+1<x1 (1 + x)^{-1/2} = 1 - \frac{1}{2}x + \frac{1\cdot 3}{2\cdot 4}x^2 - \frac{1\cdot 3 \cdot 5}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1
(1+x)1/2=1+12x124x2+13246x3+1<x1 (1 + x)^{1/2} = 1 + \frac{1}{2}x - \frac{1}{2\cdot 4}x^2 + \frac{1\cdot 3}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1

Series for exponential and logarithmic functions

ex=1+x+x22!+x33!+ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
ax=1+xlna+(xlna)22!+(xlna)33!+ a^x = 1 + x\,\ln a + \frac{(x\,\ln a)^2}{2!} + \frac{(x\,\ln a)^3}{3!} + \cdots
ln(1+x)=xx22+x33x44+1<x1 \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad -1 < x \leq 1
ln(1+x)=(x1x)+12(x1x)2+13(x1x)3+x12 \ln(1+x) = \left(\frac{x-1}{x}\right) + \frac{1}{2}\left(\frac{x-1}{x}\right)^2 + \frac{1}{3}\left(\frac{x-1}{x}\right)^3 + \cdots \quad x \geq \frac{1}{2}

Series for trigonometric functions

sinx=xx33!+x55!x77!+ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
cosx=1x22!+x44!x66!+ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
tanx=x+x33+2x515++22n(22n1)Bnx2n1(2n)!π2<x<π2 \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots + \frac{2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}
cotx=1xx3x34522nBnx2n1(2n)!0<x<π \cot x = \frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi
secx=1+x22+5x424+61x6720++Enx2n(2n)!π2<x<π2 \sec x = 1+ \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots + \frac{E_n x^{2n}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}
cscx=1x+x6+7x3360++2(22n1)Enx2n(2n)!0<x<π \csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \cdots + \frac{2\left(2^{2n}-1\right)E_n x^{2n}}{(2n)!} \quad 0 < x < \pi

Series for inverse trigonometric functions

arcsinx=x+12x33+1324x55+135246x77+1<x<1 \arcsin x = x + \frac{1}{2}\frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^7}{7} + \cdots \quad -1 < x <1
arccosx=π2arcsinx=π2(x+12x33+1324x55+)1<x<1 \arccos x = \frac{\pi}{2} - \arcsin x = \frac{\pi}{2} - \left(x + \frac{1}{2}\frac{x^3}{3} + \frac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\cdots \right) \quad -1 < x < 1
arctanx={xx33+x55x77+1<x<1π21x+13x315x5+x1π21x+13x315x5+x<1 \arctan x = \left\{ \begin{aligned} x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots & \quad -1 < x < 1 \\ \frac{\pi}{2} - \frac{1}{x} + \frac{1}{3x^3} - \frac{1}{5x^5} + \cdots & \quad x \geq 1 \\ -\frac{\pi}{2} - \frac{1}{x} + \frac{1}{3x^3} - \frac{1}{5x^5} + \cdots & \quad x < 1 \end{aligned} \right.
arccotx=π2arctanx={π2(xx33+x55+)1<x<11x13x3+15x5x1π+1x13x3+15x5x<1 \mathrm{arccot}\,x = \frac{\pi}{2} - \arctan x = \left\{ \begin{aligned} \frac{\pi}{2} - \left(x - \frac{x^3}{3} + \frac{x^5}{5} + \cdots \right) & \quad -1 < x < 1 \\ \frac{1}{x} - \frac{1}{3x^3} + \frac{1}{5x^5} - \cdots & \quad x \geq 1 \\ \pi + \frac{1}{x} - \frac{1}{3x^3} + \frac{1}{5x^5} - \cdots & \quad x < 1 \end{aligned} \right.

Series for hyperbolic functions

sinhx=x+x33!+x55!+ \sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots
coshx=1+x22!+x44!+ \cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots
tanhx=xx33+2x515+(1)n122n(22n1)Bnx2n1(2n)!+x<π2 \tanh x = x - \frac{x^3}{3} + \frac{2x^5}{15} + \cdots \frac{(-1)^{n-1} 2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} + \cdots \quad |x| < \frac{\pi}{2}
cothx=1x+x3x345+(1)n122nBnx2n1(2n)!+0<x<π \coth x = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + \cdots \frac{(-1)^{n-1} 2^{2n}B_nx^{2n-1}}{(2n)!} + \cdots \quad 0 < |x| < \pi

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