Day 07 in Mat360

Last time: Section 1.3: algorithms and convergence

Next time: Section 2.1: Bisection

Today:

Announcements:
- Homework 1.2 due Friday
- Whoops, I forgot to hand out the 2.1 material, so we'll do an optional "replacement" quiz.
Questions over 1.3?
Section 1.3:
- section 1.3 summary
- section 1.3 worksheet
- Today we want to wrap up the notion of convergence, by considering a few problems from section 1.3.
  Convergence basically means that you're getting close to something, and we're interested in characterizing the rate at which we're closing in. Let's look at a few examples. One thing that you'll notice is the utility of the Taylor series here.
  - 6c, p. 37
    
    If there exists a positive constant ${\kappa}$ such that
    ${|\alpha_n-\alpha|\le\kappa|\beta_n|}$
    for large values of n
    then we say that ${\{\alpha_n\}_{n=1}^{\infty}}$ converges to ${\alpha}$ with rate of convergence ${O(\beta_n)}$ .
  - 7d, p. 37
    
    Suppose that
    
    ${\lim_{h\to{0}}G(h)=0}$ and ${\lim_{h\to{0}}F(h)=L}$ .
    
    If ${\exists{K}>0}$ such that
    ${|F(h)-L|\le{K}|G(h)|,}$
    for sufficiently small h, then ${F(h)=L+O(G(h))}$ .

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