Chapter 11 Section 1: Sequences
Welcome to the exciting study of sequences! In this section, we'll learn about sequences, which are ordered lists of numbers. Understanding sequences is a crucial step in calculus, so let's get started!
What is a Sequence?
A sequence can be thought of as a list of numbers written in a specific order:
$$a_1, a_2, a_3, a_4, ..., a_n, ...$$- $a_1$ is called the first term.
- $a_2$ is the second term.
- $a_n$ is the nth term.
We will focus on infinite sequences, where each term $a_n$ has a successor $a_{n+1}$.
Notice that for every positive integer $n$, there is a corresponding number $a_n$. Therefore, a sequence can be defined as a function whose domain is the set of positive integers. We usually write $a_n$ instead of the function notation $f(n)$ for the value of the function at the number $n$.
The sequence ${a_1, a_2, a_3, ...}$ is also denoted by ${a_n}$ or ${a_n}_{n=1}^{\infty}$.
Examples of Sequences
Sequences can be defined by providing a formula for the $n$th term. Here are some examples:
- ${ \frac{n}{n+1} }_{n=1}^{\infty}$: Here, $a_n = \frac{n}{n+1}$. This gives us the sequence $\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ..., \frac{n}{n+1}, ...\}$.
- ${ \frac{(-1)^{n}(n+1)}{3^n} }_{n=1}^{\infty}$: Here, $a_n = \frac{(-1)^{n}(n+1)}{3^n}$. This sequence looks like $\{\frac{2}{3}, -\frac{3}{9}, \frac{4}{27}, -\frac{5}{81}, ..., \frac{(-1)^{n}(n+1)}{3^n}, ...\}$.
- ${\sqrt{n-3}}_{n=3}^{\infty}$: Here, $a_n = \sqrt{n-3}$ for $n \geq 3$. The sequence begins with ${0, 1, \sqrt{2}, \sqrt{3}, ..., \sqrt{n-3}, ...}$.
- ${\cos(\frac{n\pi}{6})}_{n=0}^{\infty}$: Here, $a_n = \cos(\frac{n\pi}{6})$ for $n \geq 0$. The sequence starts as ${1, \frac{\sqrt{3}}{2}, \frac{1}{2}, 0, ..., \cos(\frac{n\pi}{6}), ...\}$.
Convergence and Divergence
A sequence ${a_n}$ has a limit $L$ if the terms $a_n$ get arbitrarily close to $L$ as $n$ becomes large. We write this as:
$$ \lim_{n \to \infty} a_n = L $$If the limit exists, the sequence converges (or is convergent) to $L$. Otherwise, the sequence diverges (or is divergent).
Limit Laws for Sequences
If ${a_n}$ and ${b_n}$ are convergent sequences and $c$ is a constant, then the following limit laws hold:
- $\lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n$
- $\lim_{n \to \infty} (a_n - b_n) = \lim_{n \to \infty} a_n - \lim_{n \to \infty} b_n$
- $\lim_{n \to \infty} c a_n = c \lim_{n \to \infty} a_n$
- $\lim_{n \to \infty} (a_n b_n) = \lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n$
- $\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n}$, if $\lim_{n \to \infty} b_n \neq 0$
Keep Practicing!
Keep practicing with different sequences and applying these concepts. The more you practice, the better you'll become at understanding and working with sequences. Good luck, and have fun exploring the world of sequences!