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 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# 6 SERIES

## 6.1 FOURIER SERIES

1. If $f\left(x\right)$ is a bounded periodic function of period $2L$ (i.e., $f\left(x+2L\right)=f\left(x\right)$ ), and satisfies the Dirichlet conditions:
1. In any period $f\left(x\right)$ is continuous, except possibly for a finite number of jump discontinuities.
2. In any period $f\left(x\right)$ has only a finite number of maxima and minima,
then $f\left(x\right)$ may be represented by the Fourier series$f\left(x\right)=\frac{{a}_{0}}{2}+\sum _{n=1}^{\infty }\left({a}_{n}cos\frac{n\pi x}{L}+{b}_{n}sin\frac{n\pi x}{L}\right)$ where ${a}_{n}$ and ${b}_{n}$ are given below. This series will converge to $f\left(x\right)$ at every point where $f\left(x\right)$ is continuous, and to $\frac{f\left({x}^{+}\right)+f\left({x}^{-}\right)}{2}$ (i.e., the average of the left-hand and right-hand limits) at every point where $f\left(x\right)$ has a jump discontinuity. $\begin{array}{cc}{a}_{n}\hfill & =\frac{1}{L}{\int }_{-L}^{L}f\left(x\right)cos\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }n=0,\phantom{\rule{0.2em}{0ex}}1,2,3,\dots ,\hfill \\ \multicolumn{1}{c}{{b}_{n}}& =\frac{1}{L}{\int }_{-L}^{L}f\left(x\right)sin\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }n=1,2,3,\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$ We may also write ${a}_{n}=\frac{1}{L}{\int }_{\alpha }^{\alpha +2L}f\left(x\right)cos\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx$ and ${b}_{n}=\frac{1}{L}{\int }_{\alpha }^{\alpha +2L}f\left(x\right)sin\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx$ where $\alpha$ is any real number. Thus using $\alpha =0$ : $\begin{array}{cc}{a}_{n}\hfill & =\frac{1}{L}{\int }_{0}^{2L}f\left(x\right)cos\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }n=0,1,2,3,\dots ,\hfill \\ \multicolumn{1}{c}{{b}_{n}}& =\frac{1}{L}{\int }_{0}^{2L}f\left(x\right)sin\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}d,\mathrm{ }n=1,2,3,\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$
2. If in addition to the restrictions for (23), if $f\left(x\right)$ is an even function (i.e., $f\left(-x\right)=f\left(x\right)$ ), then the Fourier series reduces to $f\left(x\right)=\frac{{a}_{0}}{2}+\sum _{n=1}^{\infty }{a}_{n}cos\frac{n\pi x}{L}$ That is, ${b}_{n}=0$ . In this case, a simpler formula for ${a}_{n}$ is ${a}_{n}=\frac{2}{L}{\int }_{0}^{L}f\left(x\right)\phantom{\rule{0.2em}{0ex}}cos\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }n=0,\phantom{\rule{0.2em}{0ex}}1,2,3,\dots$
3. If in addition to the restrictions for (23), if $f\left(x\right)$ is an odd function (i.e., $f\left(-x\right)=-f\left(x\right)$ ), then the Fourier series reduces to $f\left(x\right)=\sum _{n=1}^{\infty }\phantom{\rule{0.2em}{0ex}}{b}_{n}sin\frac{n\pi x}{L}$ That is, ${a}_{n}=0$ . In this case, a simpler formula for ${b}_{n}$ is ${b}_{n}=\frac{2}{L}{\int }_{0}^{L}f\left(x\right)sin\frac{n\pi x}{L}\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }n=1,2,3,\dots$
4. Using the Euler relation ${e}^{i\theta }=cos\theta +isin\theta$ , we obtain the complex form of the Fourier series $f\left(x\right)=\frac{1}{2}\sum _{n=-\infty }^{n=+\infty }{c}_{n}{e}^{i{\omega }_{n}x}$ where ${c}_{n}=\frac{1}{L}{\int }_{-L}^{L}f\left(x\right)\phantom{\rule{0.2em}{0ex}}{e}^{-i{\omega }_{n}x}\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }n=0,±1,±2,±3,\dots$ with ${\omega }_{n}=\frac{n\pi }{L}$ for $n=0,±1,±2,\dots$ The set of coefficients ${c}_{n}$ is referred to as the Fourier spectrum.
5. If $f\left(x\right)$ has period $2L$ , if it is expandable by a Fourier series, and if both sine and cosine terms are present, then equations (26) and (27) can be written as $f\left(x\right)=\frac{{a}_{0}}{2}+\sum _{n=1}^{\infty }{c}_{n}sin\left(\frac{n\pi x}{L}+{\phi }_{n}\right),$ where ${a}_{n}={c}_{n}sin{\phi }_{n},\mathrm{ }{b}_{n}={c}_{n}cos{\phi }_{n},\mathrm{ }{c}_{n}=\sqrt{{a}_{n}^{2}+{b}_{n}^{2}},\mathrm{ }{\phi }_{n}=arctan\left(\frac{{a}_{n}}{{b}_{n}}\right)$ They can also be represented as $f\left(x\right)=\frac{{a}_{0}}{2}+\sum _{n=1}^{\infty }{c}_{n}\phantom{\rule{0.2em}{0ex}}cos\left(\frac{n\pi x}{L}+{\phi }_{n}\right),$ where ${a}_{n}={c}_{n}cos{\varphi }_{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{n}=-{c}_{n}sin{\varphi }_{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{n}=\sqrt{{a}_{n}^{2}+{b}_{n}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\varphi }_{n}=arctan\left(-\frac{{b}_{n}}{{a}_{n}}\right)$ and ${\phi }_{n}$ is chosen so as to make ${a}_{n}$ , ${b}_{n}$ , and ${c}_{n}$ hold.
6. The following table of trigonometric identities is helpful when developing Fourier series. Note that $n$ is an integer. $n$$n$ even $n$ odd $n/2$ odd $n/2$ even
$sinn\pi$ 0 0 0 0 0
$cosn\pi$ $\left(-1\right){}^{n}$ $+1$ $-1$ $+1$ $+1$
$sin\frac{n\pi }{2}$ 0 $\left(-1\right){}^{\left(n-1\right)/2}$ 0 0
$cos\frac{n\pi }{2}$ $\left(-1\right){}^{n/2}$ 0 $-1$ $+1$
$sin\frac{n\pi }{4}$ $\frac{\sqrt{2}}{2}\left(-1\right){}^{\left({n}^{2}+4n+11\right)/8}$ $\left(-1\right){}^{\left(n-2\right)/4}$ 0
Note also the useful formula for $sin\frac{n\pi }{2}$ and $cos\frac{n\pi }{2}$ : $\begin{array}{cc}sin\frac{n\pi }{2}\hfill & =\frac{\left(i\right){}^{n+1}}{2}\left[\left(-1\right){}^{n}-1\right]\hfill \\ \multicolumn{1}{c}{cos\frac{n\pi }{2}}& =\frac{\left(i\right){}^{n}}{2}\left[\left(-1\right){}^{n}+1\right]\hfill \\ \multicolumn{1}{c}{}\end{array}$

Auxiliary Formulas for Fourier Series

 1 $1=\frac{4}{\pi }\left[sin\frac{\pi x}{k}+\frac{1}{3}sin\frac{3\pi x}{k}+\frac{1}{5}sin\frac{5\pi x}{k}+\dots \right]$ $\left[0 2 $x=\frac{2k}{\pi }\left[sin\frac{\pi x}{k}-\frac{1}{2}sin\frac{2\pi x}{k}+\frac{1}{3}sin\frac{3\pi x}{k}-\dots \right]$ $\left[-k 3 $x=\frac{k}{2}-\frac{4k}{{\pi }^{2}}\left[cos\frac{\pi x}{k}+\frac{1}{{3}^{2}}cos\frac{3\pi x}{k}+\frac{1}{{5}^{2}}cos\frac{5\pi x}{k}+\dots \right]$ $\left[0 4 ${x}^{2}=\frac{2{k}^{2}}{{\pi }^{3}}\left[\left(\frac{{\pi }^{2}}{1}-\frac{4}{1}\right)sin\frac{\pi x}{k}-\frac{{\pi }^{2}}{2}sin\frac{2\pi x}{k}+\left(\frac{{\pi }^{2}}{3}-\frac{4}{{3}^{3}}\right)sin\frac{3\pi x}{k}-\frac{{\pi }^{2}}{4}sin\frac{4\pi x}{k}+\left(\frac{{\pi }^{2}}{5}-\frac{4}{{5}^{3}}\right)sin\frac{5\pi x}{k}+\cdots \right]$ $\left[0 5 ${x}^{2}=\frac{{k}^{2}}{3}-\frac{4{k}^{2}}{{\pi }^{2}}\left[cos\frac{\pi x}{k}-\frac{1}{{2}^{2}}cos\frac{2\pi x}{k}+\frac{1}{{3}^{2}}cos\frac{3\pi x}{k}-\frac{1}{{4}^{2}}cos\frac{4\pi x}{k}+\dots \right]$ $\left[-k

Fourier Expansions for Basic Periodic Functions

 1 $f\left(x\right)=\frac{4}{\pi }\sum _{n=1,3,5\dots }\frac{1}{n}sin\frac{n\pi x}{L}$ 2 $f\left(x\right)=\frac{2}{\pi }\sum _{n=1}^{\infty }\frac{\left(-1\right){}^{n}}{n}\left(cos\frac{n\pi c}{L}-1\right)sin\frac{n\pi x}{L}$ 3 $f\left(x\right)=\frac{c}{L}+\frac{2}{\pi }\sum _{n=1}^{\infty }\frac{\left(-1\right){}^{n}}{n}sin\frac{n\pi c}{L}cos\frac{n\pi x}{L}$ 4 $f\left(x\right)=\frac{2}{L}\sum _{n=1}^{\infty }sin\frac{n\pi }{2}\frac{sin\left(\frac{1}{2}n\pi c/L\right)}{\frac{1}{2}n\pi c/L}sin\frac{n\pi x}{L}$ 5 $f\left(x\right)=\frac{2}{\pi }\sum _{n=1}^{\infty }\frac{\left(-1\right){}^{n+1}}{n}sin\frac{n\pi x}{L}$ 6 $f\left(x\right)=\frac{1}{2}-\frac{4}{{\pi }^{2}}\sum _{n=1,3,5,\dots }\frac{1}{{n}^{2}}cos\frac{n\pi x}{L}$ 7 $f\left(x\right)=\frac{8}{{\pi }^{2}}\sum _{n=1,3,5,\dots }\frac{\left(-1\right){}^{\left(n-1\right)/2}}{{n}^{2}}sin\frac{n\pi x}{L}$ 8 $f\left(x\right)=\frac{1}{2}-\frac{1}{\pi }\sum _{n=1}^{\infty }\frac{1}{n}sin\frac{n\pi x}{L}$ 9 $f\left(x\right)=\frac{1}{2}\left(1+a\right)+\frac{2}{{\pi }^{2}\left(1-a\right)}\sum _{n=1}^{\infty }\frac{1}{{n}^{2}}\left[\left(-1\right){}^{n}cosn\pi a-1\right]cos\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 10 $f\left(x\right)=\frac{2}{\pi }\sum _{n=1}^{\infty }\frac{\left(-1\right){}^{n-1}}{n}\left[1+\frac{sinn\pi a}{n\pi \left(1-a\right)}\right]sin\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 11 $f\left(x\right)=\frac{1}{2}-\frac{4}{{\pi }^{2}\left(1-2a\right)}\sum _{n=1,3,5,\dots }\frac{1}{{n}^{2}}cosn\pi acos\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 12 $f\left(x\right)=\frac{2}{\pi }\sum _{n=1}^{\infty }\frac{\left(-1\right){}^{n}}{n}\left[1+\frac{1+\left(-1\right){}^{n}}{n\pi \left(1-2a\right)}sinn\pi a\right]sin\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 13 $f\left(x\right)=\frac{4}{\pi }\sum _{n=1}^{\infty }\frac{1}{n}sin\frac{n\pi }{4}sinn\pi asin\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 14 $f\left(x\right)=\frac{9}{{\pi }^{2}}\sum _{n=1}^{\infty }\frac{1}{{n}^{2}}sin\frac{n\pi }{3}sin\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 15 $f\left(x\right)=\frac{32}{3{\pi }^{2}}\sum _{n=1}^{\infty }\frac{1}{{n}^{2}}sin\frac{n\pi }{4}sin\frac{n\pi x}{L};\mathrm{ }\left(a=\frac{c}{2L}\right)$ 16 $f\left(x\right)=\frac{1}{\pi }+\frac{1}{2}sin\omega t-\frac{2}{\pi }\sum _{n=2,4,6,\dots }\frac{1}{{n}^{2}-1}cosn\omega t$

## 6.2 BINOMIAL SERIES

The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to be understood that the series converges for all finite values of $x$ .

 1 $\left(x+y\right){}^{n}={x}^{n}+{nx}^{n-1}y+\frac{n\left(n-1\right)}{2!}{x}^{n-2}{y}^{2}+\frac{n\left(n-1\right)\left(n-2\right)}{3!}{x}^{n-3}{y}^{3}+\dots$ $\left({y}^{2}<{x}^{2}\right)$ 2 $\left(1±x\right){}^{n}=1±nx+\frac{n\left(n-1\right){x}^{2}}{2!}±\frac{n\left(n-1\right)\left(n-2\right){x}^{3}}{3!}+\dots$ $\left({x}^{2}<1\right)$ 3 $\left(1±x\right){}^{-n}=1\mp nx+\frac{n\left(n+1\right){x}^{2}}{2!}\mp \frac{n\left(n+1\right)\left(n+2\right){x}^{3}}{3!}+\dots$ $\left({x}^{2}<1\right)$ 4 $\left(1±x\right){}^{-1}=1\mp x+{x}^{2}\mp {x}^{3}+{x}^{4}\mp {x}^{5}+\dots$ $\left({x}^{2}<1\right)$ 5 $\left(1±x\right){}^{-2}=1\mp 2x+3{x}^{2}\mp 4{x}^{3}+5{x}^{4}\mp 6{x}^{5}+\dots$ $\left({x}^{2}<1\right)$

## 6.3 REVERSION OF SERIES

Let a series be represented by $y={a}_{1}x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+{a}_{4}{x}^{4}+{a}_{5}{x}^{5}+{a}_{6}{x}^{6}+\dots$ with ${a}_{1}\ne 0$ . The coefficients of the series $x={A}_{1}y+{A}_{2}{y}^{2}+{A}_{3}{y}^{3}+{A}_{4}{y}^{4}+{A}_{5}{y}^{5}+\dots$ are $\begin{array}{cc}{A}_{1}\hfill & =\frac{1}{{a}_{1}}\mathrm{ }\mathrm{ }{A}_{2}=-\frac{{a}_{2}}{{a}_{1}^{3}}\mathrm{ }\mathrm{ }{A}_{3}=\frac{1}{{a}_{1}^{5}}\left(2{a}_{2}^{2}-{a}_{1}{a}_{3}\right)\mathrm{ }\mathrm{ }{A}_{4}=\frac{1}{{a}_{1}^{7}}\left(5{a}_{1}{a}_{2}{a}_{3}-{a}_{1}^{2}{a}_{4}-5{a}_{2}^{3}\right)\hfill \\ \multicolumn{1}{c}{{A}_{5}}& =\frac{1}{{a}_{1}^{9}}\left(6{a}_{1}^{2}{a}_{2}{a}_{4}+3{a}_{1}^{2}{a}_{3}^{2}+14{a}_{2}^{4}-{a}_{1}^{3}{a}_{5}-21{a}_{1}{a}_{2}^{2}{a}_{3}\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$

## 6.4 TAYLOR SERIES

1. $f\left(x\right)=f\left(a\right)+\left(x-a\right){f}^{\prime }\left(a\right)+\frac{\left(x-a\right){}^{2}}{2!}{f}^{\prime \prime }\left(a\right)+\frac{\left(x-a\right){}^{3}}{3!}{f}^{\prime \prime \prime }\left(a\right)+\dots +\frac{\left(x-a\right){}^{n}}{n!}{f}^{\left(n\right)}\left(a\right)+\dots \mathrm{ }\text{(Taylor"s Series, increment form)}$
2. $\begin{array}{lll}f\left(x+h\right)\hfill & =\hfill & f\left(x\right)+h{f}^{\prime }\left(x\right)+\frac{{h}^{2}}{2!}{f}^{″}\left(x\right)+\frac{{h}^{3}}{3!}{f}^{‴}\left(x\right)+\cdots \hfill \\ \hfill & =\hfill & f\left(h\right)+x{f}^{\prime }\left(h\right)+\frac{{x}^{2}}{2!}{f}^{″}\left(h\right)+\frac{{x}^{3}}{3!}{f}^{‴}\left(h\right)+\cdots \hfill \end{array}$
3. If $f\left(x\right)$ is a function possessing derivatives of all orders throughout the interval $\left[a,b\right]$ then the Taylor's series with remainder term is (for $a\le x\le b$ ) $f\left(x\right)=f\left(a\right)+\left(x-a\right){f}^{\prime }\left(a\right)+\frac{\left(x-a\right){}^{2}}{2!}{f}^{\prime \prime }\left(a\right)+\dots +\left(x-a\right){}^{n-1}\frac{{f}^{\left(n-1\right)}\left(a\right)}{\left(n-1\right)!}+{R}_{n}$ where ${R}_{n}=\frac{{f}^{\left(n\right)}\left[a+\theta ·\left(x-a\right)\right]}{n!}\left(x-a\right){}^{n},\mathrm{ }0<\theta <1.$ Special cases: There are values $0<\theta <1$ and $a , such that $\begin{array}{cc}f\left(b\right)\hfill & =f\left(a\right)+\left(b-a\right){f}^{\prime }\left(a\right)+\frac{\left(b-a\right){}^{2}}{2!}{f}^{\prime \prime }\left(a\right)+\dots +\frac{\left(b-a\right){}^{n-1}}{\left(n-1\right)!}{f}^{\left(n-1\right)}\left(a\right)+\frac{\left(b-a\right){}^{n}}{n!}{f}^{\left(n\right)}\left(X\right)\hfill \\ \multicolumn{1}{c}{f\left(a+h\right)}& =f\left(a\right)+{hf}^{\prime }\left(a\right)+\frac{{h}^{2}}{2!}{f}^{\prime \prime }\left(a\right)+\dots +\frac{{h}^{n-1}}{\left(n-1\right)!}{f}^{\left(n-1\right)}\left(a\right)+\frac{{h}^{n}}{n!}{f}^{\left(n\right)}\left(a+\theta h\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$
4. Taylor's series for a function of two variables: Define the notation: $\begin{array}{cc}\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)f\left(x,y\right)\hfill & =h\frac{\partial f\left(x,y\right)}{\partial x}+k\frac{\partial f\left(x,y\right)}{\partial y};\hfill \\ \multicolumn{1}{c}{{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)}^{2}f\left(x,y\right)}& ={h}^{2}\frac{\partial {}^{2}f\left(x,y\right)}{\partial {x}^{2}}+2hk\frac{\partial {}^{2}f\left(x,y\right)}{\partial x\partial y}+{k}^{2}\frac{\partial {}^{2}f\left(x,y\right)}{\partial {y}^{2}}\hfill \end{array}$ with an extnsion to higher powers. If ${{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)}^{n}f\left(x,y\right)|}_{x=a}^{y=b}$ where the bar and subscripts means that after differentiation we are to replace $x$ by $a$ and $y$ by $b$ , then $f\left(a+h,\phantom{\rule{0.2em}{0ex}}b+k\right)=f\left(a,b\right)+{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)f\left(x,y\right)|}_{x=a}^{y=b}+\dots +\frac{1}{n!}{{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)}^{n}f\left(x,y\right)|}_{x=a}^{y=b}+\dots$
5. Maclaurin series (use $a=0$ in equation (28)) $f\left(x\right)=f\left(0\right)+{xf}^{\prime }\left(0\right)+\frac{{x}^{2}}{2!}{f}^{\prime \prime }\left(0\right)+\frac{{x}^{3}}{3!}{f}^{\prime \prime \prime }\left(0\right)+\dots +{x}^{n-1}\frac{{f}^{\left(n-1\right)}\left(0\right)}{\left(n-1\right)!}+{R}_{n}$ where ${R}_{n}=\frac{{x}^{n}{f}^{\left(n\right)}\left(\theta x\right)}{n!},\mathrm{ }0<\theta <1.$

## 6.5 EXPONENTIAL SERIES

$\begin{array}{cc}e\hfill & =1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\dots \hfill \\ \multicolumn{1}{c}{{e}^{\phantom{\rule{0.2em}{0ex}}x}}& =1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}+\dots \hfill \\ \multicolumn{1}{c}{{a}^{x}}& =1+xlog{}_{e}a+\frac{\left(xlog{}_{e}a\right){}^{2}}{2!}+\frac{\left(xlog{}_{e}a\right){}^{3}}{3!}+\dots \hfill \\ \multicolumn{1}{c}{{e}^{\phantom{\rule{0.2em}{0ex}}x}}& ={e}^{a}\left[1+\left(x-a\right)+\frac{\left(x-a\right){}^{2}}{2!}+\frac{\left(x-a\right){}^{3}}{3!}+\dots \right]\hfill \\ \multicolumn{1}{c}{}\end{array}$

## 6.6 LOGARITHMIC SERIES

$\begin{array}{cccc}\hfill log{}_{e}\left(1+x\right)& \hfill =\hfill & x-\frac{1}{2}{x}^{2}+\frac{1}{3}{x}^{3}-\frac{1}{4}{x}^{4}+\dots & \hfill \left(-1

## 6.7 TRIGONOMETRIC SERIES

Let ${B}_{n}$ represent the ${n}^{\text{th}}$ Bernoulli number and let ${E}_{n}$ represent the ${n}^{\text{th}}$ Euler number.

 • $sinx=x-\frac{{x}^{3}}{3!}+\frac{{x}^{5}}{5!}-\frac{{x}^{7}}{7!}+\dots$ (all real values of $x$ ) • $cosx=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\frac{{x}^{6}}{6!}+\dots$ (all real values of $x$ ) • $tanx=x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\frac{17{x}^{7}}{315}+\frac{62{x}^{9}}{2835}+\dots +\frac{\left(-1\right){}^{n-1}{2}^{2n}\left({2}^{2n}-1\right){B}_{2n}}{\left(2n\right)!}{x}^{2n-1}+\dots$ $\left({x}^{2}<\frac{{\pi }^{2}}{4}\right)$ • $cotx=\frac{1}{x}-\frac{x}{3}-\frac{{x}^{3}}{45}-\frac{2{x}^{5}}{945}-\frac{{x}^{7}}{4725}-\dots -\frac{\left(-1\right){}^{n+1}{2}^{2n}}{\left(2n\right)!}{B}_{2n}{x}^{2n-1}-\dots$ $\left({x}^{2}<{\pi }^{2}\right)$ • $secx=1+\frac{{x}^{2}}{2}+\frac{5}{24}{x}^{4}+\frac{61}{720}{x}^{6}+\frac{277}{8064}{x}^{8}+\dots +\frac{\left(-1\right){}^{n}}{\left(2n\right)!}{E}_{2n}{x}^{2n}+\dots$ $\left({x}^{2}<\frac{{\pi }^{2}}{4}\right)$ • $cscx=\frac{1}{x}+\frac{x}{6}+\frac{7}{360}{x}^{3}+\frac{31}{15,120}{x}^{5}+\frac{127}{604,800}{x}^{7}+\dots +\frac{\left(-1\right){}^{n+1}2\left({2}^{2n-1}-1\right)}{\left(2n\right)!}{B}_{2n}{x}^{2n-1}+\dots$ $\left({x}^{2}<{\pi }^{2}\right)$ • $sin{}^{-1}x=x+\frac{{x}^{3}}{2·3}+\frac{1·3}{2·4·5}{x}^{5}+\frac{1·3·5}{2·4·6·7}{x}^{7}+\dots$ $\left({x}^{2}<1,\phantom{\rule{0.2em}{0ex}}-\frac{\pi }{2} • $cos{}^{-1}x=\frac{\pi }{2}-\left(x+\frac{{x}^{3}}{2·3}+\frac{1·3}{2·4·5}{x}^{5}+\frac{1·3·5{x}^{7}}{2·4·6·7}+\dots \right)$ $\left({x}^{2}<1,\phantom{\rule{0.2em}{0ex}}0 • $tan{}^{-1}x=x-\frac{{x}^{3}}{3}+\frac{{x}^{5}}{5}-\frac{{x}^{7}}{7}+\dots$ $\left({x}^{2}<1\right)$ • $tan{}^{-1}x=\frac{\pi }{2}-\frac{1}{x}+\frac{1}{3{x}^{3}}-\frac{1}{5{x}^{5}}+\frac{1}{7{x}^{7}}-\dots$ $\left(x>1\right)$ • $tan{}^{-1}x=-\frac{\pi }{2}-\frac{1}{x}+\frac{1}{3{x}^{3}}-\frac{1}{5{x}^{5}}+\frac{1}{7{x}^{7}}-\dots$ $\left(x<-1\right)$ • $cot{}^{-1}x=\frac{\pi }{2}-x+\frac{{x}^{3}}{3}-\frac{{x}^{5}}{5}+\frac{{x}^{7}}{7}-\dots$ ( ${x}^{2}<1$ )
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The 'Search Chemicals' page can be found by clicking the flask icon in the navigation bar at the top of this page. For more detailed information on how to use the chemical search, including adding properties, saving searches, exporting search results and more, click the help icon in to top right of this page, next to the welcome login message.

Below is an example of a chemical entry, showing its structure, physical properties and document tables in which it appears.

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Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.

## Cookies We Use and Their Purpose

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## First and Third Party Cookies

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## Questions or Concerns

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Here is a list of cookies we have defined as 'Strictly Necessary':

### Taylor and Francis 'First Party' Cookies

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Here is a list of the cookies we have defined as 'Performance'.

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Accessibility

The Voluntary Product Accessibility Template (VPAT) is a self-assessment document which discloses how accessible Information and Communication Technology products are in accordance with global standards.

The VPAT disclosure templates do not guarantee product accessibility but provide transparency around the product(s) and enables direction when accessing accessibility requirements.

Taylor & Francis has chosen to complete the International version of VPAT which encompasses Section 508 (US), EN 301 549 (EU) and WCAG2.1 (Web Content Accessibility Guidelines) for its products.