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 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.