Appreciate Some Guidance, Help, Steps on this question

Hello, jono1966!

Quote:

$\text{8.1 Use Maclaurin’s series to find the first 4 terms of the expansion of }\sinh(2x)$

$\text}8.2. Use this expansion to evaluate }\sinh(2x)\text{ when }x = \frac{1}{2}\text{, to 4 decimal places.}$

The Maclaurin series begins: . $\sinh(2x) \;=\;(2x) + \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} + \frac{(2x)^7}{7!} + \cdots$

If $x = \tfrac{1}{2}$ , we have: . $\sinh(1) \;=\;1 + \frac{1^3}{3!} + \frac{1^5}{5!} + \frac{1^7}{7!} \;=\; 1.175198413$

Therefore: . $\sinh(1) \:\approx\:1.1752$