Power Series Expansion for Hyperbolic Sine Function

Theorem

The hyperbolic sine function has the power series expansion:

$\ds \sinh x$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$

$\ds $

$=$

$\ds x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} + \cdots$

valid for all $x \in \R$.

$\dfrac \d {\d x} \sinh x = \cosh x$

$\dfrac \d {\d x} \cosh x = \sinh x$

Hence:

$\ds \dfrac {\d^2} {\d x^2} \sinh x$

$=$

$\ds \sinh x$

and so for all $m \in \N$:

$\ds m = 2 k: \ \ $

$\ds \dfrac {\d^m} {\d x^m} \sinh x$

$=$

$\ds \sinh x$

$\ds m = 2 k + 1: \ \ $

$\ds \dfrac {\d^m} {\d x^m} \sinh x$

$=$

$\ds \cosh x$

where $k \in \Z$.

This leads to the Maclaurin series expansion:

$\ds \sinh x$

$=$

$\ds \sum_{r \mathop = 0}^\infty \paren {\frac {x^{2 k} } {\paren {2 k}!} \map \sinh 0 + \frac {x^{2 k + 1} } {\paren {2 k + 1}!} \map \cosh 0}$

$\ds $

$=$

$\ds \sum_{r \mathop = 0}^\infty \frac {x^{2 k + 1} } {\paren {2 k + 1}!}$

$\map \sinh 0 = 0$, $\map \cosh 0 = 1$

From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Hyperbolic Functions: $20.33$
1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.33.$