Calculus: 10 - Power Series

Define power series

Power series generalise the geometric series.

Define radius of convergence

there can be at most one radius of convergence for a given power series

How to find the radius of convergence using limsup?

Proof of using limsup to find the radius of convergence

How to find the radius of convergence using the ratio test?

Addition of power series proposition

Define smoothness of a function

Define real analytical function

Given a function f : R → R that is smooth at 0, we can develop a power series for f with c = 0 that has a positive radius of convergence. If this power series has the same outputs as function f within that radius of convergence, then f is called a real analytical function. Not every smooth real function is analytical.

Maclaurin series for a function f that is infinitely differentiable at 0

General form of the Taylor series

Differentiation and integration of power series theorem

the power series expansion of f around x0 is…

Method to find the solution of an ODE (ordinary differential equation)