We differentiate power series term by term.

Suppose that the power series converges for all in some open interval . Then, on this interval, the power series represents a differentiable function and its derivative is given by In other words, the derivative of a power series is a power series. and the derivative is computed term by term, as we would differentiate a polynomial.

(problem 1) Find the derivatives of the following power series. Also, compare the interval of convergence of the original series to its derivative.