Question

Why is it helpful to know the Maclaurin series for a few basic functions?

Short answer

It is helpful because when we know Maclaurin series for a few basic series then it would become quite easy to find the Maclaurin series for many series related to it.

Step-by-step solution

Step 1. Given Information

The given term is Maclaurin series

Step 2. Explanation

It is helpful because when we know Maclaurin series for a few basic series then it would become quite easy to find the Maclaurin series for many series related to it.

For example, $f\left(x\right)=\sin (-5x^2)$

Maclaurin series for the function $g\left(x\right)=\sin x\mathrm{is}\sin x=\sum _{k=0}^{\infty }\frac{{(-1)}^k}{(2k+1)!}{x}^{2k+1}$

So, to find the Maclaurim series for the above function $f\left(x\right)=\sin (-5x^2)$, replace $xby-5x^2$ of the function sinx.

role="math" localid="1649317481496" $$\begin{gathered} \sin (-5x^2)=\sum _{k=0}^{\infty }\frac{{(-1)}^k}{(2k+1)!}{(-5x^2)}^{2k+1} \\ =\sum _{k=0}^{\infty }\frac{{(-1)}^k{(-1)}^{2k+1}{5}^{2k+1}}{(2k+1)!}{x}^{4K+2} \\ =\sum _{k=0}^{\infty }\frac{{(-1)}^{k+2k+1}{5}^{2k+1}}{(2k+1)!}{x}^{4K+2} \end{gathered}$$