Day 37 in MAT229

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Announcements:
- We're at the end of week 10.
  Next week we have our third exam, which will cover material through alternating series (you'll get notes on that subject today!).
- I'm going to harp on this again: know your definitions; make sure that you have the distinction between sequences and series clear.
  1. A sequence is an infinite, ordered list of numbers: $\left\{a_k\right\}_{k=1}^{\infty}$
    We can think of it as a function $a(x)$, but with integers for arguments, and real numbers for values: \[ a: \mathbb{N} \rightarrow \mathbb{R} \]
    It might be better simply to write it as $a(n)$, but mathematicians have adopted the alternative notation $a_n$, perhaps to make clear that this function has a domain restricted to integers.
  2. A series is an infinite sum of terms in a sequence. You can think of it as the analogue of an improper integral: \[ \sum_{k=1}^{\infty}a_k \]
    And this is one of the reasons that we are not too surprised when we find ourselves comparing series and improper integrals.
- We check convergence of a series $S$, a sum of the infinite sequence of terms $a_k$, by creating a new sequence $s_n$ of partial sums! It all seems rather self-referential.
  This is completely analogous to the process of checking the convergence of an improper integral, however. I'll describe that process (good review for your exam next week!), and include in parentheses the analogue for checking series:
  To check the convergence of the integral I (series S) of function $f(x)$ (sequence $a_k$) \[ I=\int_1^\infty f(x)\ dx \] e.g. $f(x)=\frac{1}{x^2}$, we construct function $g(R)$ (sequence $s_n$) \[ g(R)=\int_1^R\frac{1}{x^2}\ dx \]
  and check the convergence of function $g(R)$ (sequence $s_n$) as its argument goes to $\infty$ ($R\to\infty$ for the integral, $n\to\infty$ -- by integers only -- for the sequence).
  The primary difference is that one process occurs on the real numbers, and the other process occurs on the set of integers we call the natural numbers. To add up -- we might say "accumulate" -- real-valued functions, we use integrals; to add up sequences, we use sums.
Today: Alternating Series
- These are among the friendliest series. I hope that you'll find a nice pet at this function zoo.
- As we did for improper integrals, we find comparison with known creatures a useful approach. We look at several different approaches (including using improper integrals, via the Integral Test).
  Remember that it's all about the tail. In terms of convergence, we don't care about any handful of terms at the head of a series: \[ \sum_{k=1}^\infty a_k = \sum_{k=1}^m a_k + \sum_{k=m+1}^\infty a_k \]
- Materials for today:
  1. Alternating Series Test (.nb)
  2. Alternating Series Test (.pdf)
  3. Long talks you through
    - the AST, and
    - the AST error bound.
- Links/Notes:
  - Nothing to mention.

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