Exponential Series

The series $\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}$ is called an exponential series; it can be shown to converge for every real $x$ (unlike the binomial and logarithmic series, which need $|x|<1$).

where $e=1+\dfrac1{1!}+\dfrac1{2!}+\dfrac1{3!}+\cdots$ (the series at $x=1$). Replacing $x\to-x$ in (5.1):

In particular, $\dfrac1e=e^{-1}=1-\dfrac1{1!}+\dfrac1{2!}-\dfrac1{3!}+\cdots$.

Splitting into even/odd parts. Adding and subtracting (5.1) and (5.2) isolates the even- and odd-power terms:

At $x=1$: $\dfrac{e+e^{-1}}2=1+\dfrac1{2!}+\dfrac1{4!}+\dfrac1{6!}+\cdots$ and $\dfrac{e-e^{-1}}2=\dfrac1{1!}+\dfrac1{3!}+\dfrac1{5!}+\cdots$ — closed-form values for the pure-even and pure-odd factorial-reciprocal series.

Rescaling. Substituting $2x$ for $x$ in (5.1): …