Power series summation operator is linear

Statement

Additivity

Consider two power series:

$\sum _{k=0}^{\mathrm{\infty }}{a}_k(x-x_0{)}^k$

$\sum _{k=0}^{\mathrm{\infty }}{b}_k(x-x_0{)}^k$

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:

$\sum _{k=0}^{\mathrm{\infty }}(a_k+b_k)(x-x_0{)}^k$

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

Related facts

• Taylor series operator is linear