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.

example 1 Find the derivative of the power series This power series converges on the
interval . On that interval, its derivative is given by We can re-index to rewrite this
as We have seen that the original power series above represents the function on the
interval . The derivative of this function is Thus, we have a power series
representation for this function: