Question

derive the series representation of tanhx.

Answer 1

\[\begin{align*}\tanh x&=\frac{\sinh x}{\cosh x}\\\\
&=\frac{\dfrac{e^x-e^{-x}}{2}}{\dfrac{e^x+e^{-x}}{2}}\\\\
&=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\\\
&=\frac{\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\right)-\left(1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\cdots\right)}{\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\right)+\left(1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\cdots\right)}\\\\
&=\frac{2\left(x+\frac{x^3}{3!}+\frac{x^5}{5!}\cdots\right)}{2\left(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots\right)}\\\\
&=\frac{x+\frac{x^3}{3!}+\frac{x^5}{5!}\cdots}{1+\frac{x^2}{2!}+\frac{x^4}{4!}\cdots}
\end{align*}\]
Long division:

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