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:

It is possible that the power series representation of the derivative does not converge
at an endpoint of the interval of convergence of the original function even if
the power series representation of original function does converge at that
endpoint.

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

2024-09-27 13:57:45