Power Series Expansion for Hyperbolic Cotangent Function/Mistake

Source Work

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables

Chapter $20$: Taylor Series

Series for Hyperbolic Functions: $20.36$

Mistake

$\coth x = \dfrac 1 x + \dfrac x 3 - \dfrac {x^3} {45} + \dfrac {2 x^5} {945} + \cdots \dfrac {\paren {-1}^n 2^{2 n} B_n x^{2 n - 1} } {\paren {2 n}!} + \cdots \quad 0 < \size x < \pi$

Correction

Basically correct, but for good form:

there is a $+$ sign where it should technically be a $-$ sign

there should be a $+$ or $-$ sign either side of the $\cdots$.

Indeed, $\mathsf{Pr} \infty \mathsf{fWiki}$ would present this:

$\coth x = \dfrac 1 x + \dfrac x 3 - \dfrac {x^3} {45} + \dfrac {2 x^5} {945} - \cdots + \dfrac {\paren {-1}^n 2^{2 n} B_{2 n} x^{2 n - 1} } {\paren {2 n}!} + \cdots \quad 0 < \size x < \pi$

because the $x^7$ term is actually negative.

Also note that $B_n$ in the original page uses the archaic form of the Bernoulli numbers definition, while $\mathsf{Pr} \infty \mathsf{fWiki}$ uses the more modern $B_{2 n}$ form.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Hyperbolic Functions: $20.36$