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.
$π - \frac{π^{2}}{2!} + \frac{π^{3}}{3!} - \frac{π^{4}}{4!} + \frac{π^{5}}{5!} - \dots$

Short Answer

Expert verified

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

Step by step solution

Given information

The series is $π - \frac{π^{2}}{2!} + \frac{π^{3}}{3!} - \frac{π^{4}}{4!} + \frac{π^{5}}{5!} - \dots$

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

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

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

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

Or,

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

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

Find the values of the series.

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

Therefore,

$π - \frac{π^{2}}{2!} + \frac{π^{3}}{3!} - \frac{π^{4}}{4!} + \frac{π^{5}}{5!} - \dots = 1 - e^{- π} π - \frac{π^{2}}{2!} + \frac{π^{3}}{3!} - \frac{π^{4}}{4!} + \frac{π^{5}}{5!} - \dots = 1 - e^{- π}$

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

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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Use the Maclaurin series for sinx,cosx, and exto find the values of the following series.π-π22!+π33!-π44!+π55!-⋯

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

Logic and Functions

Mechanics Maths

Calculus

Applied Mathematics

Geometry

Probability and Statistics

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.
$π - \frac{π^{2}}{2!} + \frac{π^{3}}{3!} - \frac{π^{4}}{4!} + \frac{π^{5}}{5!} - \dots$