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

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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.

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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

Electricity and Magnetism

Dynamics

Fields in Physics

Atoms and Radioactivity

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Study anywhere. Anytime. Across all devices.

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