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Some important series of functions

On this page we are going to post some series of functions, which are used in mathematics at undergraduate level.

  • $\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} +\ldots \qquad -\infty <x< \infty$
  • $\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} +\ldots \qquad -\infty <x< \infty$
  • $\tanh x = x - \frac{x^3}{3} + 2\frac{x^5}{15} - 17\frac{x^7}{315} +\ldots \qquad \left |x\right |< \frac{\pi}{2}$
  • $\coth x = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} +\ldots \qquad 0 <x< \pi$
  • $\mathrm{sech} x = 1-\frac{x^2}{2}+\frac{5x^4}{24}-\frac{61x^6}{720} +\ldots \qquad \left |x\right |<\frac{\pi}{2}$
  • $\mathrm{csch} x = \frac{1}{x}-\frac{x}{6}+\frac{7x^3}{360} -\ldots \qquad 0<\left |x\right |<\pi$
  • $\sinh^{-1} x = \pm{ln\left |2x\right |+\frac{1}{2.2x^2}-\frac{1.3}{2.4.4x^4} +\ldots}$ + if $x\geq 1$, - if $x\leq 1$
  • $\cosh^{-1 }x = \pm{ln(2x)-(\frac{1}{2.2x^2}+\frac{1.3}{2.4.4x^4} +. . .}$
  • $\tanh^{-1} x = x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7} +. . . \qquad \left |x\right |<1$
  • $\coth^{-1} x = \frac{1}{x}+\frac{1}{3x^3}+\frac{1}{5x^5}+\frac{1}{7x^7} +\ldots \qquad \left |x\right |>1$


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