Chapter 8: Q. 18 (page 692)

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!}$

Short Answer

Expert verified

The required answer is $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!} = - 1$

Step by step solution

Step 1. Given Information

The given series is $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!}$

Step 2. Explanation

The Maclaurin series for the function $f (x) = \cos x is \cos x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} x^{2 k}}{(2 k)!}$

So, the series $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!}$ is the maclaurin series for $\cos x at x = π$

Since, $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} x^{2 k}}{(2 k)!} = \cos x$

Thus,

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!} = cosπ \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!} = - 1$

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.∑k=0∞-1kπ2k2k!

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} π^{2 k}}{(2 k)!}$