Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

#28 Monday (11/25/96)

SUMMARY Power Series

READING §23

To find a power series of f(x) in Derive, use Calculus, Taylor. Then input the function, the variable, and the order of the series (highest power in the polynomial). As output, you will get something like: TAYLOR(f(x), x, 0, n) (Note: You can just Author the Taylor command yourself rather than going through the menu.) The 0 represents the point about which you took the power series. Let’s deal with only 0 right now. You can do it about any point you wish. To actually see the Taylor series written out, you must use Expand (or Simplify, or approX). Examples. Find the power series about x = 0 for the following functions:

(a) e^{x} =

(b) sin(x) =

(c) \frac{1}{1 − x} =

(d) ln(1 − x) =

Let’s now look at the plots of some of these functions versus the first few terms of the their Taylor series. Plot sin(x), TAYLOR(sin(x), x, 0, 3), TAYLOR(sin(x), x, 0, 5), TAYLOR(sin(x), x, 0, 9) on the same plot. Make some observations.

Plot ln(1−x), TAYLOR(ln(1-x), x, 0, 3), TAYLOR(ln(1-x), x, 0, 5), TAYLOR(ln(1-x), x, 0,

9) on the same plot. Make some observations and contrast with those for the sine function.

Sometimes we have infinite power series, but really don’t know what functions they converge to. Look at the examples in # 23.1 on page 132. Use the Calculus, Sum commands to define the partial sums of some of these series. Then plot a few of the partial sums for the same series on the same plot. What can you say about the “intervals of convergence”?

Power Series Given a sequence (a_{n})_{0}^{∞}, the series ∑_{n=0}^{∞} a_{n}x^{n} is called a POWER SERIES.

As we have seen, the power series is a function of x. It converges for x = 0. By convention, we take 0^{0} = 1. Given it any sequence (a_{n}), ∑ a_{n}x^{n} either

• CONVERGES for all x ∈ ℝ

• CONVERGES for only x = 0

• CONVERGES for all x in some bounded interval

Radius Of Convergence 23.1 For the power series ∑ a_{n}x^{n}, let β = lim sup |a_{n}|^{1/n} and R = 1/β (If β = 0 → R = +∞ but if β = +∞ → R = 0) The power series either

• CONVERGES for |x| < R

• DIVERGES for |x| > R

For |x| = R (i.e. x = −R and x = R) convergence of the power series must be tested separately.

R is called the radius of convergence.

PROOF CASE 1: 0 < R < ∞

CASE 2: R = +∞

CASE 3: R = 0

Remember: even though lim sup|a_{n}|^{1/n} can be difficult to compute, if lim_{n→∞} |\frac{a_{n+1}}{a_{n}} | exists and

equals L, then

L = lim |a_{n}|^{1/n} = lim sup |a_{n}|^{1/n}

Homework Assignment (Due Monday December 2) §23, page 132: 23.1, 23.4