Question

$\mathit{s}\mathit{i}\mathit{n}{\mathit{x}}^{\mathbf{2}}$

Short answer

The series is $\sin \left(x^2\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{4n+2}}{(2n+1)!}$

The sum is given below.

$S_1=x^2$

$S_2=x^2-\frac{x^6}{3!}$

$S_3=x^2-\frac{x^6}{3!}+\frac{x^{10}}{5!}$

The graphs are shown below.

Step-by-step solution

Given Information

The Maclaurin series.

Definition of the MaclaurinSeries.

A Maclaurin series is a function with an expansion series that gives the sum of the function's derivatives.

Find the series and sum.

The Maclaurin series is given below

$\sin \left(x\right)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots$

$\sin \left(x^2\right)=x^2-\frac{x^6}{3!}+\frac{x^{10}}{5!}+\dots$

The general series is given below.

$\sin \left(x\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$

$\sin \left(x^2\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{4n+2}}{(2n+1)!}$

The sum is given below

$S_1=x^2$

$S_2=x^2-\frac{x^6}{3!}$

$S_3=x^2-\frac{x^6}{3!}+\frac{x^{10}}{5!}$

Plot the graphs.

The graph of the function with $S_1,S_2,S_3$is given below.