Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.

using

Solution 33E

Step 1:

In this problem we need to find the power series representation for g(x) = centered at ‘0’ by differentiating or integrating the power series for f(x) =

We already know a power series for f(x) = = for

, since , and =

Therefore , power series representation for g(x) =

Step 2:

Now let us find the interval of convergence of

Use Ratio test to determine the interval of convergence.

If there exists an N so that for all , and and

If L<1, converges.If L>1, diverges.If L=1, the test is inconclusive.We have

, since

, since as n , then

Step 3:

The series converges for L < 1

Therefore solve |x| < 1

By using , “ if |f(x)| < a then -a < f(x) < a “ we get -1 < x < 1