The hyperbolic sine and hyperbolic cosine functions, denoted...

The hyperbolic sine and hyperbolic cosine functions, denoted sinh and cosh respectively, are defined as $$\sinh x=\frac{e^{x}-e^{-x}}{2} \text { and } \cosh x=\frac{e^{x}+e^{-x}}{2}$$ (Appendix A6 gives more information about hyperbolic functions.) Find the Maclaurin series generated by $\sinh x$ and $\cosh x .$

The hyperbolic sine and hyperbolic cosine functions, denoted sinh and cosh respectively, are defined as
$$\sinh x=\frac{e^{x}-e^{-x}}{2} \text { and } \cosh x=\frac{e^{x}+e^{-x}}{2}$$
(Appendix A6 gives more information about hyperbolic functions.)
Find the Maclaurin series generated by $\sinh x$ and $\cosh x .$

Key Concepts

Maclaurin Series

A Maclaurin series is a type of Taylor series expansion of a function about the point 0. It expresses a function as an infinite sum of its derivatives evaluated at zero, each multiplied by powers of the variable divided by factorial terms. This concept is fundamental in approximating functions and analyzing their behavior near the origin.

Exponential Function Series

The exponential function e^x can be represented as an infinite series: the sum from n = 0 to infinity of x^n/n!. This series expansion is crucial because it provides a basis for deriving the series for other functions, such as the hyperbolic sine and cosine, by substituting x with appropriate expressions like x and -x.

Hyperbolic Functions

Hyperbolic functions, such as sinh x and cosh x, are defined using exponential functions. The hyperbolic sine, sinh x, is defined as (e^x - e^(-x))/2, and the hyperbolic cosine, cosh x, as (e^x + e^(-x))/2. Their definitions mirror the trigonometric sine and cosine functions but are built from exponential functions, leading to their distinctive growth properties and series representations.

Parity of Functions (Even and Odd Functions)

The parity properties of functions help simplify their series expansions. A function is even if f(-x) = f(x) and odd if f(-x) = -f(x). In the context of hyperbolic functions, cosh x is even (only even-powered terms appear) and sinh x is odd (only odd-powered terms appear), which greatly simplifies their Maclaurin series representations.

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