The power series is .

The series converges for .

Find the radius of the convergence of .

Differentiation and integration of power series.

Theorem 2:

If the power series has radius of convergence then the function is differentiable on the interval   and

(i) , (ii)    then 

The radii of the power series in (i) and (ii) are both .

Here is the integral of the sum .

Therefore by theorem 2, series converges for .

Radius of the convergence of is also .

Radius of the convergence of is .