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Chapter 1: Q20p (page 32)

$e^{x} s i n x$

Short Answer

Expert verified

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

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

Step by step solution

01

Given Information.

The given function.

$e^{x} \sin x$

02

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 $0 .$

03

Find the Maclaurin series for exsinx..

Use the Maclaurin series of $\sin (x)$ and $e^{x}$

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

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

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

Further, multiply $\sin (x)$ and $e^{x}$

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

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

04

Use MATLAB tool to find Maclaurin series.

Find first few terms of Maclaurin series.

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

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

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Most popular questions from this chapter

$\int_{0}^{x} e^{- t^{2}} d t$

$x^{2} l n (1 - x)$

Solve for all possible values of the real numbersand in the following equations.

$\frac{x + i y + 2 + 3 i}{2 x + 2 i y - 3} = i + 2$

In the following problems, find the limit of the given sequence as $n \to \infty$

$\frac{10^{n}}{n!}$

Test the following series for convergence

$\sum_{n = 0}^{^{\infty}} \frac{(- 1)^{n} n}{1 + n^{2}}$

See all solutions

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