Power Series Lesson 9.8

Definition A power series centered at 0 has the form Each is a fixed constant The coefficient of

Example A geometric power series Consider for which real numbers x does S(x) converge? Try x = 1, x = ½ Converges for |x| < 1 Limit is

An e Example It is a fact … (later we see why) The right side is a power series We seek the values of x for which the series converges

An e Example We use the ratio test Thus since 0 < 1, series converges absolutely for all values of x Try evaluating S(10), S(20), S(30)

Power Series and Polynomials Consider that power series are polynomials Unending Infinite-degree The terms are power functions Partial sums are ordinary polynomials

Choosing Base Points Consider These all represent the same function Try expanding them Each uses different base point Can be applied to power series

Choosing Base Points Given power series Written in powers of x and (x – 1) Respective base points are 0 and 1 Note the second is shift to right We usually treat power series based at x = 0

Definition A power series centered at c has the form This is also as an extension of a polynomial in x

Examples Where are these centered, what is the base point?

Power Series as a Function Domain is set of all x for which the power series converges Will always converge at center c Otherwise domain could be An interval (c – R, c + R) All reals c c

Example Consider Think of S(x) as a geometric series What is the domain? Think of S(x) as a geometric series a = 1 r = 2x Geometric series converges for |r| < 1

Finding Interval of Convergence Often the ratio test is sufficient Consider Show it converges for x in (-1, 1)

Finding Interval of Convergence Ratio test As k gets large, ratio tends to |x| Thus for |x| < 1 the series is convergent

Convergence of Power Series For the power series centered at c exactly one of the following is true The series converges only for x = c There exists a real number R > 0 such that the series converges absolutely for |x – c| < R and diverges for |x – c| > R The series converges absolutely for all x

Example Consider the power series What happens at x = 0? Use generalized ratio test for x ≠ 0 Try this

Dealing with Endpoints Consider Converges trivially at x = 0 Use ratio test Limit = | x | … converges when | x | < 1 Interval of convergence -1 < x < 1

Dealing with Endpoints Now what about when x = ± 1 ? At x = 1, diverges by the divergence test At x = -1, also diverges by divergence test Final conclusion, convergence set is (-1, 1)

Try Another Consider Again use ratio test Should get which must be < 1 or -1 < x < 5 Now check the endpoints, -1 and 5

Power Assignment Lesson 9.8 Page Exercises 1 – 33 EOO