Power Series A power series is an infinite polynomial. We can generalize by centering the power series at x = c.

These are functions of x. There may be no simpler way to express these functions.

A power series is convergent at a value of x if the infinite sum converges to a finite number when evaluated at x. The interval of convergence is the interval of x values that make the series converge. The radius of convergence is the distance away from x = c that we can go to get convergence. Radius = ∞ Interval =  Radius = 0  Interval = just the point c

Ex. Find the radius of convergence for

Ex. Find the radius of convergence for

Ex. Find the interval of convergence and radius of convergence for

Ex. Find the interval of convergence and radius of convergence for

We can take the derivative and integral of the power series term by term  The radius of convergence won’t change, though the endpoints of the interval might

Ex. For the function , find the interval of convergence for f ʹ(x).

Representing Functions as Power Series Recall the geometric series: We can use this to write some function as a power series. So as long as |x| < 1.

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R = 3, (-5,1) Pract. 1. Find the ROC and IOC for 2. Find the power series for R = 3, (-5,1)