Derivative of power series

Definition

Power series centered at a point

The derivative of a power series centered at a point $\mu$ is the power series obtained by diferentiating each term of the power series and adding up. Explicitly, the derivative of the power series:

$\sum _{k=0}^{\infty }a_{k}(x-\mu )^{k}$

is the power series

$\sum _{k=1}^{\infty }ka_{k}(x-\mu )^{k-1}$

Rewriting the dummy variable $k$ as $k+1$, we can rewrite the above summation as:

$\sum _{k=0}^{\infty }(k+1)a_{k+1}(x-\mu )^{k}$

Facts

• Differentiation theorem for power series: This states that if a power series converges to a function on a given interval of convergence, the derivative of the power series converges to the derivative of the function on the interior of that interval of convergence. It may happen, though, that the power series converges to the function at the boundary of the interval of convergence but the power series for the derivative does not converge.
• Taylor series operator commutes with differentiation: The Taylor series of the derivative of a function at a point is the derivative (in the sense of derivative of power series) of the Taylor series.