Problem 5E

Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.

Solution 5E

Step 1:

Consider the function ,

which is the derivative of the function

This has the power series representation of the form

This series converges when .

The power series representation of is obtained by differentiating the series term by term.thus we get

This also converges when .

Thus the interval and radius of convergence of a power series does not change on differentiating.