Day 10 in Mat228

Last time: More sequences

Next time: More Series

Today:

Announcements
- Quiz on Series
- Homework for 9.4 returned. Comments from the grader:
  1. #40: Most weren't sure how to prove for all n or k. Only a few caught the pattern in the derivatives.
  2. #46: Most didn't get that they should expand sin(x) first then plug in ${x=\theta/2}$ and ${x=\theta/4}$ .
  3. #51: It seems that many were too intimidated and I quote: "Yup ... my brain just fried."
  4. Almost everyone seemed to do well overall.
  5. I had several who wrote their Taylor polynomials about a and then expanded them: e.g. ${\frac{1}{2}+\frac{1}{2}(x+1)+\frac{1}{4}(x+1)^2=\frac{x^2}{4}+x+\frac{5}{4}}$
- Homework on sequences is due today.
- Your exam (Thursday, next week) will cover through 11.1. There are hence only four sections covered: 8.1, 8.6, 9.4, and 11.1.
- Your homework on series will be due after the first exam.
Section 11.1: Sequences
- Examples:
  - #47, p. 544. Hint:
  - The sequence of the Fibonacci numbers and some Fibonacci fun...
Section 11.2: Series
- A handout concerning series.
- Once we have some series in hand, we might want to start operating with them. In particular, as we saw in our very first series, we might want to add up the terms:
  
  In particular, we expect that, if we throw in the ${0^{th}}$ term, we should get
  ${\sum_{n=1}^{\infty}\frac{5^n}{n!}=e^5-1\approx{147.4}}$
  Such as sum is called an infinite series.
- The question which presents itself is this: what do we mean by this sum of an infinite number of terms? This is indeed curious.
- Here's how we make sense of that infinite sum:
  - From the sequence ${\{a_n=\frac{5^n}{n!}\}}$
    we define a new sequence ${\{s_k\}}$ :
    ${\{s_k=\sum_{n=1}^{k}a_n=\sum_{n=1}^{k}\frac{5^n}{n!}\}}$
    These are called partial sums. If
    
    ${\lim_{k \to \infty}s_k={\lim_{k \to \infty}\sum_{n=1}^{k}\frac{5^n}{n!}=L}$
    Then we say that
    ${\sum_{n=1}^{\infty}\frac{5^n}{n!}}$
    exists, and is equal to L. More formally,
    
    Now geometric series are sufficiently important that it's useful to include that special case:
    
    There's a really fun proof of the first part above, based on a trick. Let's have a look at it.
    
    This theorem tells us that if a sequence isn't asymptotic to the x-axis, we can forget about its partial sums converging.
    Finally, series behave the way we'd hope:
  - Examples:
    - #7, p. 554
    - #8, p. 554 (compare to Taylor series error bound)
    - Reflect on #36, p. 555. Recursive actions and definitions give rise to these types of geometric sums.
    - Along the same lines: #45, p. 555
    - #52, p. 556
    - Beautiful spiral "proofs" of some infinite series

Website maintained by Andy Long. Comments appreciated.