Power Series Section 9.1a

Definition: Infinite Series A finite sum of real numbers always produces a real number, but an infinite sum of real numbers is not actually a real sum: Definition: Infinite Series An infinite series is an expression of the form or The numbers are the terms of the series; is the nth term.

A finite sum of real numbers always produces a real number, but an infinite sum of real numbers is not actually a real sum: The partial sums of the series form a sequence If the sequence of partial sums has a limit S as n approaches infinity, we say the series converges to S: Otherwise, we say the series diverges.

Guided Practice Does the given series diverge? Can we use the associate property of addition?: This does NOT work with infinite series!!! Instead, look at the sequence of partial sums: Since this sequence has no finite limit, then the original series has no sum. It diverges.

Guided Practice Does the given series diverge? Look at the sequence of partial sums: This sequence has a limit , which we recognize as 1/3. The series converges to the sum 1/3. This last problem is an example of a geometric series, because each term is obtained by multiplying the previous term by the same constant (1/10 in this case)…

Geometric Series The geometric series converges to the sum if , and diverges if The interval is the interval of convergence. (these are the r values for which the series converges)

Guided Practice Tell whether each series converges or diverges. If it converges, give its sum. The first term is a = 3 and r = 1/2. The series converges to: The first term is a = 1 and r = –1/2. The series converges to:

Guided Practice Tell whether each series converges or diverges. If it converges, give its sum. The first term is a = 9/25 and r = 3/5. The series converges to: In this series, The series diverges.

Representing Functions by Series If , then the geometric series formula gives: The expression on the left defines a function whose domain is the interval of convergence, . The expression on the right defines a function whose domain is the set of all real numbers . The equality is understood to hold only in this latter domain, where both sides of the equation are defined. On this domain, the series represents the function 1/(1 – x).

Representing Functions by Series If , then the geometric series formula gives: Let’s graph the function together with some partial sums of the series. Graph in [–4.7, 4.7] by [–2, 4]: The graphs behave similarly, but only on the interval of convergence!!!

Definition: Power Series An expression of the form is a power series centered at x = 0. An expression of the form is a power series centered at x = a. The term is the nth term; the number a is the center.

Guided Practice Given that 1/(1 – x) is represented by the power series on the interval (–1, 1), find a power series that represents 1. on 2. on 3. on

Guided Practice Given that 1/(1 – x) is represented by the power series on the interval (–1, 1), find a power series that represents 4. and give its interval of convergence This geometric series converges for: Interval of convergence:

Guided Practice Given that 1/(1 – x) is represented by the power series on the interval (–1, 1), find a power series that represents 5. and give its interval of convergence This geometric series converges for: Interval of convergence: