General Taylor Expansion
$$f(a+x) = f(a) + xf'(a) + \frac{x^2}{2!}f''(a) + \cdots + \frac{x^n}{n!}f^{(n)}(a) + \cdots$$
Exponential and Logarithmic
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty}\frac{x^n}{n!} \qquad \text{(all } x\text{)}$$
$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \qquad (-1 < x \le 1)$$
$$\ln\frac{1+x}{1-x} = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots\right) \qquad (|x| < 1)$$
Binomial
$$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots \qquad (|x| < 1 \text{ for non-integer } n)$$
Terminates after \(n+1\) terms when \(n\) is a positive integer (valid for all \(x\)). Special cases:
$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \qquad (|x| < 1)$$
$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots \qquad (|x| < 1)$$
Trigonometric
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \qquad \text{(all } x\text{)}$$
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \qquad \text{(all } x\text{)}$$
$$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots \qquad \left(|x| < \tfrac{\pi}{2}\right)$$
Inverse Trigonometric
$$\arcsin x = x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{15x^7}{336} + \cdots \qquad (|x| < 1)$$
$$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \qquad (|x| \le 1)$$
Hyperbolic
$$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots \qquad \text{(all } x\text{)}$$
$$\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots \qquad \text{(all } x\text{)}$$
$$\tanh x = x - \frac{x^3}{3} + \frac{2x^5}{15} - \frac{17x^7}{315} + \cdots \qquad \left(|x| < \tfrac{\pi}{2}\right)$$
The hyperbolic series mirror their trigonometric counterparts with all positive signs (\(\tanh\) alternates where \(\tan\) does not).
Gregory's Series
$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots$$
The \(\arctan\) series at \(x=1\). Converges slowly; in practice paired with the Machin identity:
$$\frac{\pi}{4} = 4\arctan\frac{1}{5} - \arctan\frac{1}{239}$$
Secant Series (Euler Numbers)
$$\sec x = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots = \sum_{n=0}^{\infty}\frac{|E_{2n}|\,x^{2n}}{(2n)!} \qquad \left(|x| < \tfrac{\pi}{2}\right)$$
Euler numbers: \(E_0 = 1\), \(|E_2| = 1\), \(|E_4| = 5\), \(|E_6| = 61\), \(|E_8| = 1385\).
$$\operatorname{sech} x = 1 - \frac{x^2}{2} + \frac{5x^4}{24} - \frac{61x^6}{720} + \cdots \qquad \left(|x| < \tfrac{\pi}{2}\right)$$
Bernoulli-Number Forms
Bernoulli numbers: \(B_2 = \tfrac{1}{6}\), \(B_4 = -\tfrac{1}{30}\), \(B_6 = \tfrac{1}{42}\), \(B_8 = -\tfrac{1}{30}\).
$$\tan x = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}\,2^{2n}(2^{2n}-1)\,B_{2n}}{(2n)!}x^{2n-1} = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \frac{62x^9}{2835} + \cdots \qquad \left(|x| < \tfrac{\pi}{2}\right)$$
$$\cot x = \frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} - \frac{2x^5}{945} - \frac{x^7}{4725} - \cdots = \frac{1}{x} - \sum_{n=1}^{\infty}\frac{2^{2n}|B_{2n}|}{(2n)!}x^{2n-1} \qquad (0 < |x| < \pi)$$
$$\coth x = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} - \cdots = \frac{1}{x} + \sum_{n=1}^{\infty}\frac{2^{2n}B_{2n}}{(2n)!}x^{2n-1} \qquad (0 < |x| < \pi)$$
$$\tanh x = \sum_{n=1}^{\infty}\frac{2^{2n}(2^{2n}-1)\,B_{2n}}{(2n)!}x^{2n-1} = x - \frac{x^3}{3} + \frac{2x^5}{15} - \frac{17x^7}{315} + \cdots \qquad \left(|x| < \tfrac{\pi}{2}\right)$$
Related Series
$$\csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \frac{31x^5}{15120} + \cdots \qquad (0 < |x| < \pi)$$
$$\frac{x}{e^x - 1} = 1 - \frac{x}{2} + \frac{x^2}{12} - \frac{x^4}{720} + \cdots = \sum_{n=0}^{\infty}\frac{B_n x^n}{n!} \qquad (|x| < 2\pi)$$
The last expansion is the generating function defining the Bernoulli numbers. Note that \(\cot\), \(\coth\), and \(\csc\) are Laurent series rather than true power series owing to the \(\tfrac{1}{x}\) term, though handbooks tabulate them together.
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