Chapter 8: Q 28. (page 704)

Manipulating power series: Find power series representations
for the indicated functions.
$e - \frac{e^{3}}{3!} + \frac{e^{5}}{5!} - \frac{e^{7}}{7!} + \dots$

Short Answer

Expert verified

The power series representations for the indicated functions

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

Step by step solution

Given information

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

Find the values of the series.

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

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

Therefore,

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

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Manipulating power series: Find power series representations
for the indicated functions.
$e - \frac{e^{3}}{3!} + \frac{e^{5}}{5!} - \frac{e^{7}}{7!} + \dots$

Short Answer

Step by step solution

Given information

Find the values of the series.

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Manipulating power series: Find power series representationsfor the indicated functions.e-e33!+e55!-e77!+⋯

Short Answer

Step by step solution

Given information

Find the values of the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Probability and Statistics

Geometry

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Manipulating power series: Find power series representations
for the indicated functions.
$e - \frac{e^{3}}{3!} + \frac{e^{5}}{5!} - \frac{e^{7}}{7!} + \dots$