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

#29 Monday (12/02/96)

SUMMARY Power Series READING §23 \underline{Examples} Consider the following series and determine the interval of convergence for each.

∑_{n=1}^{∞} x^{n}

∑_{n=1}^{∞} n!x^{n}

∑_{n=1}^{∞} \frac{x^{n}}{n^{2}}

∑_{n=1}^{∞} \frac{(−1)^{n+1}}{n} (x − 1)^{n}

∑_{n=0}^{∞} 2^{−n}x^{3n}

\underline{Continuity} When a power series CONVERGES, it converges to a function f(x). The question is, what are the properties of this function, and how is it related to the properties of each of the partial sums ∑_{k=0}^{n} a_{k}x^{k} = f_{n}, which are also functions. (What kind of functions are they?)

Is the limit of a series of continuous functions continuous?

\underline{Example} Consider f_{n} = (1 − |x|)^{n}.

Are the f_{n} continuous? What is f(x) = lim f_{n}? Is f(x) continuous?

We call the usual type of convergence we have been talking about when lim f_{n}(x) = f(x) POINTWISE CONVERGENCE Write down the definition of this type of convergence.

There is another type of convergence, called UNIFORM CONVERGENCE Write down the definition of UNIFORM CONVERGENCE.

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