Define hyperbolic functions and the comparison test for infinite series.

Here are the solutions to the questions:

QUESTION ONE

(a) Define/State the following:

(i) Hyperbolic functions
Hyperbolic functions are a set of functions mathematically analogous to the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. They are typically defined in terms of the exponential function, for example, $\sinh x = \frac{e^x - e^{-x}}{2}$ and $\cosh x = \frac{e^x + e^{-x}}{2}$.

(ii) Comparison Test
The Comparison Test is a method used to determine the convergence or divergence of an infinite series or an improper integral by comparing it with another series or integral whose convergence or divergence is already known. For example, if $0 \le a_