Q 69. Find the values of x for which t... [FREE SOLUTION]

Chapter 7: Q 69. (page 615)

Find the values of x for which the series ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ converges.

Short Answer

Expert verified

The series ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ converges for all real numbers except for $x 鈭� \{(2 n + 1) \frac{蟺}{2} | n 鈭� 鈩�\}$ .

Step by step solution

Step 1. Given information.

Given a series ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ .

Step 2. Find all values of x for which the series converges.

A geometric series is of the form ${鈭�}_{k = 0}^{鈭�} c r^{k}$ for some constants c and r.

Supposer is a non-zero real number, then ${鈭�}_{k = 0}^{鈭�} c r^{k}$ converges to $\frac{c}{1 - r}$ if and only if $|r| < 1$ .

Here, the series localid="1648832395698" ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ has $r = \sin x$ .

For the series to converge, $|\sin x| < 1$ .

Note that $- 1 鈮� \sin x 鈮� 1$ .

So, $|\sin x| = 1$ if and only if localid="1648832424318" $x = (2 n + 1) \frac{蟺}{2}$ for all natural number n.

It follows that ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ converges for all real number except for $x = (2 n + 1) \frac{蟺}{2}$ where localid="1648832538629" $n 鈭� 鈩�$ .

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the values of x for which the series ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find all values of x for which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Find the values of x for which the series 鈭�K=0鈭�sinxkconverges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find all values of x for which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

Find the values of x for which the series ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ converges.