Power series summation operator is linear

Statement

Additivity

Consider two power series:

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

$\sum _{k=0}^{\infty }b_{k}(x-x_{0})^{k}$

Let $f$ be the function to which the first power series converges (wherever it converges) and $g$ be the function to which the second power series converges (wherever it converges). Then, the sum of the power series:

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

converges to $f+g$ at all points where both the power series converge.

Related facts

• Taylor series operator is linear