Welcome to Chapter 11! This week represents a significant shift in our calculus journey. We are moving away from continuous functions and integrals to explore the discrete world of Infinite Sequences and Series. This topic is fundamental to understanding how calculators compute functions like sine, cosine, and exponents, and it lays the groundwork for Taylor Series later in the chapter.

11.1 Sequences

A sequence is simply a list of numbers written in a definite order. We denote a sequence as $\{a_n\}$. You can think of it as a function whose domain is the set of positive integers.

The main question we ask about a sequence is: Does it approach a specific number?

  • If $\lim_{n \to \infty} a_n = L$ (a finite number), we say the sequence converges to $L$.
  • If the limit does not exist or approaches infinity, the sequence diverges.

An important tool we discussed is the Monotonic Sequence Theorem, which states that every sequence that is both bounded (contained within a range) and monotonic (constantly increasing or decreasing) must be convergent.

11.2 Infinite Series

While a sequence is a list, a series is the sum of that list. We write this using sigma notation:

$$ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots $$

It seems counterintuitive that adding an infinite amount of numbers could result in a finite sum. To check this, we look at the Sequence of Partial Sums, denoted as $s_n$. If the partial sums approach a limit $s$, then the series converges to that sum.

Key Tests for Convergence

In this lesson, we introduced two critical tests for determining convergence:

1. The Geometric Series
This is one of the few series where we can easily find the exact sum. A geometric series has the form $a + ar + ar^2 + \dots$. It converges only if the absolute value of the ratio $|r| < 1$. The sum is given by the formula:

$$ \sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r} $$

If $|r| \ge 1$, the series diverges.

2. The Test for Divergence
This is a quick check, but be careful with the logic! It states:

$$ \text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum a_n \text{ diverges.} $$

Important Note: If the limit is zero, the series might still diverge! The classic example is the Harmonic Series ($\sum \frac{1}{n}$), which diverges even though its terms get smaller and smaller.

Please review the class notes and the video lecture below to see specific examples of these tests in action. Getting comfortable with the difference between a sequence converging and a series converging is the key to success in Chapter 11!

Class Materials:
You can find the PowerPoint presentation, PDF notes, and the Zoom recording link attached below.