Question

$e^{x}sinx$

Short answer

The Maclaurin series, i.e.,$e^x\sin \left(x\right)=x+x^2+\frac{x^3}{3}-\frac{x^5}{30}+\dots$

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

Step-by-step solution

Given Information.

The given function.

$e^x\sin x$

Definition of Maclaurin series.

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 exsinx..

Use the Maclaurin series of $\sin \left(x\right)$and $e^x$

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

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$

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

Further, multiply $\sin \left(x\right)$and $e^x$

$e^x\sin \left(x\right)=x-\frac{x^3}{3!}+\frac{x^5}{5!}+x^2-\frac{x^4}{3!}+\frac{x^3}{2!}-\frac{x^5}{2!3!}+\frac{x^4}{3!}+\frac{x^5}{4!}+\dots$

$e^x\sin \left(x\right)=x+x^2+\frac{x^3}{3}-\frac{x^5}{30}+\dots$

Use MATLAB tool to find Maclaurin series.

Find first few terms of Maclaurin series.

Hence, the Maclaurin series, i.e., $e^x\sin \left(x\right)=x+x^2+\frac{x^3}{3}-\frac{x^5}{30}+\dots$

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