Power Series Expansion for Exponential of Tangent of x

Theorem

 * $e^{\tan x} = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 - \dfrac {3 x^4} 8 + \cdots$

for all $x \in \R$ such that $\left\lvert{x}\right\rvert < \frac \pi 2$.

Proof
Let $f \left({x}\right) = e^{\tan x}$.

Then:

By definition of Taylor series:


 * $f \left({x}\right) \sim \displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

This is to be expanded about $\xi = 0$.

Note that $\tan 0 = 0$ and $\sec 0 = 1$.

Thus:

from which the result.