Q. 13 - Calculus

Question

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

Step-by-Step Solution

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 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 $x by -5x^2$ of the function sinx.

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