Theory and Examples
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.
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.