# Theory and ExamplesThe series converges to sin x for all ## Problem 55E Chapter 10.7

Thomas' Calculus: Early Transcendentals | 13th Edition

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Problem 55E

Theory and Examples

The series converges to sin x for all x.

a. Find the first six terms of a series for cos x. For what values of x should the series converge?

b. By replacing x by 2x in the series for sin x, find a series that converges to sin 2x for all x.

c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for 2 sin x cos x. Compare your answer with the answer in part (b).

Step-by-Step Solution:

Step 1 of 5</p>

Here, we are given a series of that converges to for all x. We are asked to do the following.

Step 2 of 5</p>Here, we have to find the first 6 terms of the series for and we also have to find the value of x for which the series converge.

Given that We know that the derivative of is .

Therefore, we get the series for by differentiating the given series for .

Therefore, we get  Step 3 of 5</p>

Now, we have to find the value of x for which converges.

We see that, by using the ratio test, we get .

Therefore, the series converges absolutely for all values of x.

Step 4 of 5

Step 5 of 5

##### ISBN: 9780321884077

This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13th. Since the solution to 55E from 10.7 chapter was answered, more than 221 students have viewed the full step-by-step answer. Thomas' Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321884077. This full solution covers the following key subjects: Series, sin, Answer, terms, converges. This expansive textbook survival guide covers 138 chapters, and 9198 solutions. The answer to “Theory and ExamplesThe series converges to sin x for all x.a. Find the first six terms of a series for cos x. For what values of x should the series converge?b. By replacing x by 2x in the series for sin x, find a series that converges to sin 2x for all x.c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for 2 sin x cos x. Compare your answer with the answer in part (b).” is broken down into a number of easy to follow steps, and 85 words. The full step-by-step solution to problem: 55E from chapter: 10.7 was answered by , our top Calculus solution expert on 08/01/17, 02:37PM.

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Theory and ExamplesThe series converges to sin x for all

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