Power Series Expansion for Real Area Hyperbolic Cotangent

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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

From Power Series Expansion for Real Area Hyperbolic Tangent:

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

for $\size x < 1$.


From Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent:

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


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.

$\blacksquare$


Also see


Sources

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