The power series is .

Radius of the convergence of the series is .

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 derivative of the sum .

Therefore by theorem 2, radius of the convergence of is also .

Radius of the convergence of is .