Q 26 Use the Maclaurin series for sin... [FREE SOLUTION]

Chapter 8: Q 26 (page 704)

Use the Maclaurin series for $\sin x$ ,cosx, and $e^{x}$ to find the values of the following series.
$蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰�$

Short Answer

Expert verified

The values of the series $蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰�$ is $蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰� = 1 - e^{- 蟺}$

Step by step solution

Given information

The series is $蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰�$

Find the Maclaurin series for the function f(x)=ex

The Maclaurin series for the function $f (x) = e^{x}$ is

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

So, $e^{- x} = 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \frac{x^{4}}{4!} - 鈰� - e^{- x} = - 1 + x - \frac{x^{2}}{2!} + \frac{x^{3}}{3!} - \frac{x^{4}}{4!} + 鈰�$

Or,

$1 - e^{- x} = x - \frac{x^{2}}{2!} + \frac{x^{3}}{3!} - \frac{x^{4}}{4!} + 鈰�$

The series $蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰�$ is the Maclaurin series for $1 - e^{- x}$ at $x = 蟺$

Find the values of the series.

$x - \frac{x^{2}}{2!} + \frac{x^{3}}{3!} - \frac{x^{4}}{4!} + 鈰� = 1 - e^{- x}$

Therefore,

$蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰� = 1 - e^{- 蟺} 蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰� = 1 - e^{- 蟺}$

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91影视

Use the Maclaurin series for $\sin x$ ,cosx, and $e^{x}$ to find the values of the following series.
$蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰�$

Short Answer

Step by step solution

Given information

Find the Maclaurin series for the function f(x)=ex

Find the values of the series.

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Most popular questions from this chapter

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91影视

Use the Maclaurin series for sinx,cosx, and exto find the values of the following series.蟺-蟺22!+蟺33!-蟺44!+蟺55!-鈰�

Short Answer

Step by step solution

Given information

Find the Maclaurin series for the function f(x)=ex

Find the values of the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for $\sin x$ ,cosx, and $e^{x}$ to find the values of the following series.
$蟺 - \frac{蟺^{2}}{2!} + \frac{蟺^{3}}{3!} - \frac{蟺^{4}}{4!} + \frac{蟺^{5}}{5!} - 鈰�$