Day 25 in MAT229

Last time: Section 11.1

Next time: Section 11.2

Today:

Announcements
- 9.1 homework (by hand) returned:
  - #26 - why not |f(x)| in the denominator?
- 9.4 homework (Taylor Series lab) is due.
Section 11.1: Sequences
- Geometric sequence: a sequence of the form
  
  ${\{a_n=cr^n\}_{n\in\{0,1,\ldots.\}}$
  where c is a constant and r is called the common ratio.
- Examples:
  - #19, p. 543
  - #28(a), p. 544
  - #45, p. 544
  - #47, p. 544. Hint:
  - The sequence of the Fibonacci numbers and some Fibonacci fun...
Section 11.2: 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
  ${\sum_{n=0}^{\infty}\frac{5^n}{n!}=e^5\approx{148.4}}$
  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:
    ${\{a_k=\sum_{n=0}^{k}\frac{5^n}{n!}\}}$
    These are called partial sums. If
    
    ${\lim_{k \to \infty}a_k={\lim_{k \to \infty}\sum_{n=0}^{k}\frac{5^n}{n!}=L}$
    Then we say that
    ${\sum_{n=0}^{\infty}\frac{5^n}{n!}}$
    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. 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:
    - #1, p. 553
    - #7, p. 554
    - #8, p. 554 (compare to Taylor series error bound)
    - #36, p. 555
    - #45, p. 555

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