# Power Series Expansion for Real Area Hyperbolic Cotangent

## Theorem

The (real) area hyperbolic cotangent function has a Taylor series expansion:

 $\ds \arcoth x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1} x^{2 n + 1} }$ $\ds$ $=$ $\ds \frac 1 x + \frac 1 {3 x^3} + \frac 1 {5 x^5} + \frac 1 {7 x^7} + \cdots$

for $\size x > 1$.

## Proof

 $\ds \artanh x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}$ $\ds$ $=$ $\ds x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots$

for $\size x < 1$.

$\map \artanh {\dfrac 1 x} = \arcoth x$

So:

 $\ds \arcoth x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \times \frac 1 {x^{2 n + 1} }$ $\ds$ $=$ $\ds \frac 1 x + \frac 1 3 \times \frac 1 {x^3} + \frac 1 5 \times \frac 1 {x^5} + \frac 1 7 \times \frac 1 {x^7} + \cdots$

Hence the result.

$\blacksquare$