Section 11-1: Infinite Sequences
Welcome to another exciting math class! Today, we're diving into the concept of infinite sequences. An infinite sequence is essentially an ordered list of numbers that continues indefinitely. We denote a sequence as ${a_1, a_2, a_3, ..., a_n, ...}$ where $a_1$ is the first term, $a_2$ is the second term, and $a_n$ is the $n^{th}$ term.
Understanding Sequences
Let's explore some examples to solidify our understanding:
- Example 1: The sequence of reciprocals of powers of 2: ${ \{ \frac{1}{2^n} \} } = \{ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, ..., \frac{1}{2^n}, ... \}$
- Example 2: A sequence derived from the decimal places of the number $e$: If $a_n$ is the digit in the $n^{th}$ decimal place of $e$, then the sequence's first few terms are ${ \{7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5,...\} }$.
- Example 3: A sequence relating consecutive integers: ${\left\{ \frac{n}{n+1} \right\}_{n=1}^{\infty} = \left\{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ... \right\} }$
The Fibonacci Sequence
A particularly interesting sequence is the Fibonacci sequence, denoted as ${ \{ f_n \} }$. It is defined recursively as follows:
- $f_1 = 1$
- $f_2 = 1$
- $f_n = f_{n-1} + f_{n-2}$ for $n \ge 3$
This yields the sequence: ${ \{1, 1, 2, 3, 5, 8, 13, 21, ...\} }$.
Convergence and Divergence
A crucial concept when working with sequences is whether they converge or diverge.
A sequence ${ \{a_n\} }$ has the limit $L$ if, as $n$ approaches infinity, $a_n$ gets arbitrarily close to $L$. We write this as:
$$ \lim_{n \to \infty} a_n = L $$Or, equivalently, $a_n \to L$ as $n \to \infty$. If this limit $L$ exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).
Limit Laws
Several laws help us calculate limits of sequences:
- Product Law: $$\lim_{n \to \infty} (a_n b_n) = \lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n$$
- Quotient Law: $$\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n}$$, if $$\lim_{n \to \infty} b_n \neq 0$$
- Power Law: $$\lim_{n \to \infty} a_n^p = \left[ \lim_{n \to \infty} a_n \right]^p$$, if $p > 0$ and $a_n > 0$
Squeeze Theorem for Sequences
The Squeeze Theorem is a powerful tool for finding limits. If $a_n \le b_n \le c_n$ for $n \ge n_0$ and $$\lim_{n \to \infty} a_n = L = \lim_{n \to \infty} c_n$$, then $$\lim_{n \to \infty} b_n = L$$.
Examples of Convergence and Divergence
Let's work through a couple of examples:
- Find $$\lim_{n \to \infty} \frac{n}{n+1}$$. We can divide both numerator and denominator by $n$ to get: $$\lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}} = \frac{1}{1 + 0} = 1$$ Therefore, the sequence converges to 1.
- Is the sequence $a_n = \frac{n}{\sqrt{10 + n}}$ convergent or divergent? We can rewrite this as $$\lim_{n \to \infty} \frac{1}{\sqrt{\frac{10}{n^2} + \frac{1}{n}}} = \frac{1}{0} = \infty$$. Thus, the sequence is divergent.
Monotonic and Bounded Sequences
- A sequence ${ \{a_n\} }$ is called increasing if $a_n < a_{n+1}$ for all $n \ge 1$.
- A sequence ${ \{a_n\} }$ is called decreasing if $a_n > a_{n+1}$ for all $n \ge 1$.
- A sequence is called monotonic if it is either increasing or decreasing.
- A sequence ${ \{a_n\} }$ is bounded above if there is a number $M$ such that $a_n \le M$ for all $n \ge 1$.
- A sequence ${ \{a_n\} }$ is bounded below if there is a number $m$ such that $m \le a_n$ for all $n \ge 1$.
Key Theorem: Every bounded, monotonic sequence is convergent. This is a powerful result that helps us determine convergence without explicitly finding the limit!
Keep practicing, and you'll master these concepts in no time! Good luck with your studies!