Intro to Analysis in-class_Part_51

# Intro to Analysis in-class_Part_51 - 7.2 Numerical Series...

This preview shows pages 1–2. Sign up to view the full content.

7.2 Numerical Series and Sequences 107 Fix ε > 0. Since b n converges, choose N such that n N = ⇒ | β n | < ε, so that | e n | ≤ | a 0 β n + ...a n - N - 1 β N +1 | + | β N a n - N + ··· + a n β 0 | εα + | β N a n - N + ··· + a n β 0 | . Now since a n 0, for N ﬁxed and n >> 1, we can make | β N a n - N + ··· + a n β 0 | < ε . Then | e n | < ( α + 1) ε . § 7.2 Exercise: # Recommended: # 1. a n converges = a n 0. 2. If { x n } is Cauchy in R and some subsequence { x n k } converges to x R , then prove the full sequence { x n } also converges to x . 3. If a n +1 a n < r for n >> 1, then n p | a n | < r for n >> 1. 4. If lim | a n | | b n | = 1, then X | a n | converges ⇐⇒ X | a n | converges .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
108 Math 413 Sequences and Series of Functions 7.3 Uniform convergence What does it mean to say a sequence of functions { f n } converges? I.e., how to deﬁne when lim f n ( x ) = f ( x )? There are diﬀerent (nonequivalent) ways to deﬁne such a limit. What does it mean to say a sum of functions { f n } converges? I.e., how to deﬁne f n ( x ) = f ( x )? For example, power series have f n ( x ) = a n x n . What about other kinds of functions? We want to know when things are valid, like for f ( x ) = f n ( x ), f 0 ( x )? = X f 0 n ( x ) Z f ( x ) dx ? = X Z f n ( x ) dx, The Gamma function is deﬁned Γ(
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

### Page1 / 2

Intro to Analysis in-class_Part_51 - 7.2 Numerical Series...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online