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 $\size x < \dfrac \pi 2$.

Proof
Let $\map f x = e^{\tan x}$.

Then:

By definition of Taylor series:


 * $\ds \map f x \sim \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$

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

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

Thus:

Hence: