Taylor, Binomial, Trigonometric Functions, Miscellaneous, Power Series

Taylor Series for Functions of one Variable

$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^2}{2!}+\cdots+\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!}+R_n$ where $R_n$, the remainder after $n$ terms, is given by either of the following forms:

Lagrange’s form $R_n=\frac{f^{(n)}(\xi)(x-a)^n}{n!}$

Cauchy’s form $R_n=\frac{f^{(n)}(\xi)(x-\xi)^{n-1}(x-a)}{(n-1)!}$

The value $\xi$, which may be different in the two forms, lies between $a$ and $x$. The result holds if $f(x)$ has continuous derivatives of order $n$ at least.

If $\lim_{n\rightarrow\infty} R_n=0$, the infinite series obtained is called the Taylor series for $f(x)$ about $x=a$. If $a=0$ the series is often called a Maclaurin series. These series, often called power series, generally converge for all values of $x$ in some interval called the interval of convergence and diverge for all $x$ outside that interval.

Binomial Series

$(a+x)^n=a^n+na^{n-1}x+\frac{n(n-1)}{2!}a^{n-2}x^2+\frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+\cdots=$
$= a^n+\binom{n}{1}a^{n-1}x+\binom{n}{2}a^{n-2}x^2+\binom{n}{3}a^{n-3}x^3+\cdots$

Secial cases are

$(a+x)^2=a^2+2ax+x^2$

$(a+x)^3=a^3+3a^2x+3ax^2+x^3$

$(a+x)^4=a^4+4a^3x+6a^2x^2+4ax^3+x^4$

$(1+x)^{-1}=1-x+x^2-x^3+x^4-\cdots,$     $-1< x< 1$

$(1+x)^{-2}=1-2x+3x^2-4x^3+5x^4-\cdots,$     $-1< x< 1$

$(1+x)^{-3}=1-3x+6x^2-10x^3+15x^4-\cdots$     $-1< x< 1$

$(1+x)^{-\frac{1}{2}}=1-\frac{1}{2}x+\frac{1\cdot3}{2\cdot4}x^2-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}x^3+\cdots$     $-1< x\leq1$

$(1+x)^{\frac{1}{2}}=1+\frac{1}{2}x-\frac{1}{2\cdot4}x^2+\frac{1\cdot3}{2\cdot4\cdot6}x^3-\cdots$     $-1< x\leq1$

$(1+x)^{-\frac{1}{3}}=1-\frac{1}{3}x+\frac{1\cdot4}{3\cdot6}x^2-\frac{1\cdot4\cdot7}{3\cdot6\cdot9}x^3+\cdots$     $-1< x\leq1$

$(1+x)^{\frac{1}{3}}=1+\frac{1}{3}x-\frac{2}{3\cdot6}x^2+\frac{2\cdot5}{3\cdot6\cdot9}x^3-\cdots$     $-1< x\leq1$

Series if Exponential and Logarithmic Functions

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$     $-\infty< x< \infty$

$a^x=e^{x\ln a}=1+x\ln a+\frac{(x\ln a)^2}{2!}+\frac{(x\ln a)^3}{3!}+\cdots$     $ -\infty< x< \infty$

$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$    $-1< x\leq1$

$\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)=x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots$    $-1< x< 1$

$\ln x=2\left\{\left(\frac{x-1}{x+1}\right)+\frac{1}{3}\left(\frac{x-1}{x+1}\right)^3+\frac{1}{5}\left(\frac{x-1}{x+1}\right)^5+\cdots\right\}$     $x>0$

$\ln x=\left(\frac{x-1}{x}\right)+\frac{1}{2}\left(\frac{x-1}{x}\right)^2+\frac{1}{3}\left(\frac{x-1}{x}\right)^3+\cdots$     $x\geq\frac{1}{2}$

Series of Trigonometric Functions

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$     $-\infty< x< \infty$

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$     $-\infty< x< \infty$

$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+\cdots+\frac{2^{2n}(2^{2n}-1)B_nx^{2n-1}}{(2n)!}+\cdots$     $|x|< \frac{\pi}{2}$

$\cot x=\frac{1}{x}-\frac{x}{3}-\frac{x^3}{45}-\frac{2x^5}{945}-\cdots-\frac{2^{2n}B_nx^{2n-1}}{(2n)!}-\cdots$     $0< |x|< \pi$

$\sec x=1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+\cdots+\frac{E_nx^{2n}}{(2n)!}+\cdots$     $|x|< \frac{\pi}{2}$

$\csc x=\frac{1}{x}+\frac{x}{6}+\frac{7x^3}{360}+\frac{31x^5}{15,120}+\cdots+\frac{2(2^{2n-1}-1)B_nx^{2n-1}}{(2n)!}+\cdots$     $0< |x|< \pi$

$\sin^{-1}x=x+\frac{1}{2}\frac{x^3}{3}+\frac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^7}{7}+\cdots$     $|x|< 1$

$\cos^{-1}x=\frac{\pi}{2}-\sin^{-1}x=\frac{\pi}{2}-\left(x+\frac{1}{2}\frac{x^3}{3}+\frac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\cdots\right)$     $|x|< 1$

$\tan^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots$   when $|x|< 1$
$\tan^{-1}x=\pm\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^3}-\frac{1}{5x^5}+\cdots$     $[+\ \text{if}\ x\geq1, -\ \text{if}\ x\leq-1]$

$\cot^{-1}x=\frac{\pi}{2}-\tan^{-1}x =\frac{\pi}{2}-\left(x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots\right)$when $|x|< 1$
$\cot^{-1}x=\frac{\pi}{2}-\tan^{-1}x =p\pi+\frac{1}{x}-\frac{1}{3x^3}+\frac{1}{5x^5}-\cdots$ when $[p=0\ \text{if}\ x>1, p=1\ \text{if}\ x< -1]$

$\sec^{-1}x=\cos^{-1}\left(\frac{1}{x}\right)=\frac{\pi}{2}-\left(\frac{1}{x}+\frac{1}{2\cdot3x^3}+\frac{1\cdot3}{2\cdot4\cdot5x^5}+\cdots\right)$     $|x|>1$

$\csc^{-1}x=\sin^{-1}\left(\frac{1}{x}\right)=\frac{1}{x}+\frac{1}{2\cdot3x^3}+\frac{1\cdot3}{2\cdot4\cdot5x^5}+\cdots$     $|x|>1$

Series of Hyperbolic Functions

$\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots$     $-\infty< x< \infty$

$\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots$     $-\infty< x< \infty$

$\tanh x=x-\frac{x^3}{3}+\frac{2x^5}{15}-\frac{17x^7}{315}+\cdots+\frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_nx^{2n-1}}{(2n)!}+\cdots$     $|x|< \frac{\pi}{2}$

$\coth x=\frac{1}{x}+\frac{x}{3}-\frac{x^3}{45}+\frac{2x^5}{945}+\cdots+\frac{(-1)^{n-1}2^{2n}B_nx^{2n-1}}{(2n)!}+\cdots$     $0< |x|< \pi$

$\text{sech}x=1-\frac{x^2}{2}+\frac{5x^4}{24}-\frac{61x^6}{720}+\cdots+\frac{(-1)^nE_nx^{2n}}{(2n)!}+\cdots$    $|x|< \frac{\pi}{2}$

$\text{csch}x=\frac{1}{x}-\frac{x}{6}+\frac{7x^3}{360}-\frac{31x^5}{15,120}+\cdots+\frac{(-1)^n2(2^{2n-1}-1)B_nx^{2n-1}}{(2n)!}+\cdots$     $0< |x|< \pi$

$\sinh^{-1}x= x-\frac{x^3}{2\cdot3}+\frac{1\cdot3x^5}{2\cdot4\cdot5}-\frac{1\cdot3\cdot5x^7}{2\cdot4\cdot6\cdot7}+\cdots$   $|x|< 1$
$\sinh^{-1}x=\pm\left(\ln|2x|+\frac{1}{2\cdot2x^2}-\frac{1\cdot3}{2\cdot4\cdot4x^4}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot6x^6}-\cdots\right)$     $\left[+\ \text{if}\ x\geq1\quad -\ \text{if}\ x\leq-1\right]$

$\cosh^{-1}x=\pm\left\{\ln(2x)-\left(\frac{1}{2\cdot2x^2}+\frac{1\cdot3}{2\cdot4\cdot4x^4}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot6x^6}+\cdots\right)\right\}$   $\left[+\ \text{if}\ \cosh^{-1}x>0, x\geq1\quad -\ \text{if}\ \cosh^{-1}x< 0, x\geq1\right]$

$\tanh^{-1}x=x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots$     $|x|< 1$

$\coth^{-1}x=\frac{1}{x}+\frac{1}{3x^3}+\frac{1}{5x^5}+\frac{1}{7x^7}+\cdots$     $|x|>1$

Miscellaneous Series

$e^{\sin x}=1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}+\cdots$     $-\infty< x< \infty$

$e^{\cos x}=e\left(1-\frac{x^2}{2}+\frac{x^4}{6}-\frac{31x^6}{720}+\cdots\right)$     $-\infty< x< \infty$

$e^{\tan x}=1+x+\frac{x^2}{2}+\frac{x^3}{2}+\frac{3x^4}{8}+\cdots$     $|x|< \frac{\pi}{2}$

$e^x\sin x=x+x^2+\frac{2x^3}{3}-\frac{x^5}{30}-\frac{x^6}{90}+\cdots+\frac{2^\frac{n}{2}\sin\left(\frac{n\pi}{4}\right)x^n}{n!}+\cdots$     $-\infty< x< \infty$

$e^x\cos x=1+x-\frac{x^3}{3}-\frac{x^4}{6}+\cdots+\frac{2^\frac{n}{2}\cos\left(\frac{n\pi}{4}\right)x^n}{n!}+\cdots$     $-\infty< x< \infty$

$\ln|\sin x|=\ln|x|-\frac{x^2}{6}-\frac{x^4}{180}-\frac{x^6}{2835}-\cdots-\frac{2^{2n-1}B_nx^{2n}}{n(2n)!}+\cdots \qquad 0< |x|< \pi$

$\ln|\cos x|=-\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{45}-\frac{17x^8}{2520}-\cdots-\frac{2^{2n-1}(2^{2n}-1)B_nx^{2n}}{n(2n)!}+\cdots$     $|x|< \frac{\pi}{2}$

$\ln|\tan x|=\ln|x|+\frac{x^2}{3}+\frac{7x^4}{90}+\frac{62x^6}{2835}+\cdots+\frac{2^{2n}(2^{2n-1}-1)B_nx^{2n}}{n(2n)!}+\cdots$     $0< |x|< \frac{\pi}{2}$

$\frac{\ln(1+x)}{1+x}=x-\left(1+\frac{1}{2}\right)x^2+\left(1+\frac{1}{2}+\frac{1}{3}\right)x^3-\cdots$     $|x|< 1$

Reversion of Power Series

If
$y=c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5+\cdots$
then
$x=C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+\cdots$
where

$c_1C_1=1$

$c_1^3C_2=-c_2$

$c_1^5C_3=2c_2^2-c_1c_3$

$c_1^7C_4=5c_1c_2c_3-5c_2^3-c_1^2c_4$

$c_1^9C_5=6c_1^2c_2c_4+3c_1^2c_3^2-c_1^3c_5+14c_2^4-21c_1c_2^2c_3$

$c_1^{11}C_6=7c_1^3c_2c_5+84c_1c_2^3c_3+7c_1^3c_3c_4-28c_1^2c_2c_3^2-c_1^4c_6-28c_1^2c_2^2c_4-42c_2^5$

Taylor Series for Functions of Two Variables

$f(x,y)=f(a,b)+(x-a)f_x(a,b)+(y-b)f_y(a,b)+$ $\frac{1}{2!}\left\{(x-a)^2f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^2f_{yy}(a,b)\right\}+\cdots$
where $f_x(a,b), f_y(a,b),\cdots$ denote partial derivatives with respect to $x, y,\cdots$ evaluated at $x=a, y=b$.