Chapter 8: Q 27 (page 704)

Use the Maclaurin series for $\sin x$ ,
$\cos x$ , and $e^{x}$ to find the values of the following series.
$1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots$

Short Answer

Expert verified

The values of the series $1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots$ is $1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots = 0$

Step by step solution

Given information

The series is $1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots$

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

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

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

The series $1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots$ is the Maclaurin series for

$\cos x$ at $x = \frac{π}{2}$

Find the values of the series

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

Therefore,

$1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots = \cos \frac{π}{2} 1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots = 0$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the Maclaurin series for $\sin x$ ,
$\cos x$ , and $e^{x}$ to find the values of the following series.
$1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots$

Short Answer

Step by step solution

Given information

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

Find the values of the series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

Mechanics Maths

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for sinx,cosx, and exto find the values of the following series.1-π222·2!+π424·4!-π626·6!+⋯

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

Logic and Functions

Pure Maths

Probability and Statistics

Mechanics Maths

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for $\sin x$ ,
$\cos x$ , and $e^{x}$ to find the values of the following series.
$1 - \frac{π^{2}}{2^{2} \cdot 2!} + \frac{π^{4}}{2^{4} \cdot 4!} - \frac{π^{6}}{2^{6} \cdot 6!} + \dots$