Expansion of Series

Series Expansion

Polynomial Expansion: (1 + x)^a
Exponential Expansion: e^x
Sine Expansion: sin(x)
Cosine Expansion: cos(x)
Tangent Expansion: tan(x)
Hyperbolic Sine Expansion: sinh(x)
Hyperbolic Cosine Expansion: cosh(x)
Natural Logarithm Expansion: ln(1 + x)

Polynomial Expansion Series

$(1 + x)^{a} = 1 + ax + \frac{a(a-1)}{2!}x^{2} + \frac{a(a-1)(a-2)}{3!}x^{3}+...\Rightarrow (where \left | x \right |<1)$

Exponential Expansion Series

$e^{x} = 1 + x + \frac{x^{2}}{2!}+...+\frac{x^{n}}{n!}+... \Rightarrow (where \left | x \right | <\infty )$

Sine Expansion Series

$sin(x) = x - \frac{x^{3}}{3!}+\frac{x^{5}}{5!}-...+(-1)^{n}\frac{x^{2n+1}}{(2n+1)!} \Rightarrow (\textup{where }\left | x \right | < \infty )$

Cosine Expansion Series

$cos(x) = 1 - \frac{x^{2}}{2!}+ \frac{x^{4}}{4!} -... + (-1)^{n} \frac{x^{2n}}{(2n)!} + ...\Rightarrow (where \left | x \right | <\infty )$

Tangent Expansion Series

$tan(x) = x + \frac{x^{3}}{3!}+ \frac{2x^{5}}{15!} + \frac{17x^{7}}{315!} + ...\Rightarrow (\textup{where }\frac{-\pi }{2}<x<\frac{\pi }{2} )$

Hyperbolic Sine Expansion Series

$sinh(x) = \frac{e^{x}-e^{-x}}{2} = x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \frac{x^{7}}{7!} + ... \Rightarrow (\textup{where } \left | x \right | < \infty )$

Hyperbolic Cosine Expansion Series

$cosh(x) = \frac{e^{x}+e^{-x}}{2} = x + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + ... (\textup{where } \left | x \right | < \infty )$

Natural Logarithm Expansion Series

$ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - ... + (-1)^{n}\frac{x^{n+1}}{(n+1)} +... \Rightarrow (\textup{where }-1 < x \leq 1)$