Chapter 1: Q23MP (page 1)

Use the series you know to show that:
$\frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots = 1$

Short Answer

Expert verified

$\frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots . . is equal to 1$

Step by step solution

Maclaurin series and the given series

The series is $\frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots \dots$

The Maclaurin series ofis expressed as follows:

$s i n x = x - \frac{x^{3}}{6} + \frac{x^{5}}{120} + \dots \dots$

Use the Maclaurin series.

Divide the series $\sin x$ by $x$ as follows:

$(\frac{\sin x}{x}) = \frac{x}{x} - \frac{x^{3}}{6 x} + \frac{x^{5}}{120 x} + \dots \dots . (\frac{\sin x}{x}) = 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots \dots .$

Subtract $(\frac{\sin x}{x})$ from $1$ on both sides in the above equation as follows:

$1 - (\frac{\sin x}{x}) = \frac{x^{2}}{6} - \frac{x^{4}}{120} + \frac{x^{6}}{5040} \dots \dots$

Substitute $π$ for $x$ in above expansion as follows:

$\begin{array}{rcl} 1 - (\frac{\sin π}{π}) & = & \frac{π^{2}}{6} - \frac{π^{4}}{120} + \frac{π^{6}}{5040} \dots \dots \\ 1 - (\frac{0}{π}) & = & \frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots . \\ \frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots . . & = & 1 \end{array}$

Thus, $\frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots . . = 1$

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 Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the series you know to show that:
$\frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots = 1$

Short Answer

Step by step solution

Maclaurin series and the given series

Use the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Modern Physics

Further Mechanics and Thermal Physics

Mechanics and Materials

Particle Model of Matter

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Use the series you know to show that:π23!-π45!+π67!-…=1

Short Answer

Step by step solution

Maclaurin series and the given series

Use the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Modern Physics

Further Mechanics and Thermal Physics

Mechanics and Materials

Particle Model of Matter

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Use the series you know to show that:
$\frac{π^{2}}{3!} - \frac{π^{4}}{5!} + \frac{π^{6}}{7!} - \dots = 1$