Question

$\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\sqrt{\mathbf{x}}\mathbf{\right)}}{\sqrt{\mathbf{x}}}\mathbf{,}\mathit{x}\mathbf{>}\mathbf{0}$

Short answer

The Maclaurin series, i.e $\frac{\sin \left(\sqrt{x}\right)}{\sqrt{x}}=1-\frac{x}{3!}+\frac{x^2}{5!}-\frac{x^3}{7!}+\dots$

The general term, i.e. $\frac{\sin \left(\sqrt{\mathrm{x}}\right)}{\sqrt{\mathrm{x}}}=\sum _{\mathrm{n}=0}^{\infty }\frac{(-1{)}^{\mathrm{n}}\mathrm{x}}{(2\mathrm{n}+1)!}$

The Maclaurin series using MATLAB tool, i.e.,

Graph of the function and $S_1$ together, i.e.,

Graph of the function and $S_2$ together, i.e.,

Graph of the function and $S_3$ together, i.e.,

Step-by-step solution

Given Information.

The given function.

$\frac{\sin \left(\sqrt{x}\right)}{\sqrt{x}}$

Definition of Maclaurinseries.

All of the terms in a Maclaurin series are nonnegative integer powers of the variable. It's a Taylor series expansion of a function with a value of around 0.

Find the Maclaurin series for sin(x)x

Use the Maclaurin series of $\sin \left(x\right)$and replace $x$by $\sqrt{x}$,

$f\left(x\right)=f\left(0\right)+\frac{f^{\text{'}}\left(0\right)}{1!}\left(x\right)+\frac{f^{\text{'}\text{'}}\left(0\right)}{2!}\left(x^2\right)+\frac{f^{\text{'}\text{'}\text{'}}\left(0\right)}{3!}\left(x^3\right)+\dots$

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

$\sin \left(\sqrt{x}\right)=x^{1/2}-\frac{x^{3/2}}{3!}+\frac{x^{5/2}}{5!}-\frac{x^{7/2}}{7!}+\dots$

$\frac{\sin \left(\sqrt{x}\right)}{\sqrt{x}}=1-\frac{x}{3!}+\frac{x^2}{5!}-\frac{x^3}{7!}+\dots$

Write the general term of sin(x)x

Use the general term of $\sin \left(x\right)$and replace $x$by $\sqrt{x}$

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

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

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

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

Use MATLAB tool to find Maclaurin series.

Find first few terms of Maclaurin series.

Approximate partial sums of the series.

Write partial sums of the series $\left[S_1,S_2,S_3\right]$

$S_1=1-\frac{x}{6}$

$S_2=1-\frac{x}{6}+\frac{x^2}{120}$

$S_3=1-\frac{x}{6}+\frac{x^2}{120}-\frac{x^3}{5040}$

Plot the function sin(x)x and S1 together

Graph of the function and $S_1$together are shown as:

Plot the function  sin(x)x and S2 together

Graph of the function and $S_2$together are shown as:

Plot the function sin(x)xand S3 together.

Graph of the function and $S_3$together are shown as:

Hence, the Maclaurin series, i.e. $\frac{\sin \left(\sqrt{x}\right)}{\sqrt{x}}=1-\frac{x}{3!}+\frac{x^2}{5!}-\frac{x^3}{7!}+\dots$

The general term, i.e. $\frac{\sin \left(\sqrt{x}\right)}{\sqrt{x}}=\sum _{n=0}^{\infty }\frac{(-1{)}^{n}x}{(2n+1)!}$.

The Maclaurin series using MATLAB tool, i.e.,

Graph of the function and $S_1$together, i.e.,

Graph of the function and $S_2$together, i.e.,

Graph of the function and $S_3$together, i.e.,