Chapter 8: Q 25 (page 704)

Use the Maclaurin series for $\cos x, \sin x$ and $e^{x}$ to find the values of the following series.
$π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots$

Short Answer

Expert verified

The values of the series $π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots$ is $π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots = 0$

Step by step solution

Given information

The series is $π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots$

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

The Maclaurin series for the function $f (x) = \sin x$ is

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

The series $π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots$ is the Maclaurin series for $\sin x$ at $x = π$

Find the values of the series.

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

Therefore,

$π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots = \sin π π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots = 0$

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Use the Maclaurin series for $\cos x, \sin x$ and $e^{x}$ to find the values of the following series.
$π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots$

Short Answer

Step by step solution

Given information

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

Find the values of the series.

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Use the Maclaurin series for cosx,sinxand ex to find the values of the following series.π-π33!+π55!-π77!+⋯

Short Answer

Step by step solution

Given information

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

Find the values of the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for $\cos x, \sin x$ and $e^{x}$ to find the values of the following series.
$π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots$