Welcome back to Professor Baker's Math Class! In this lesson, we are shifting gears to explore Section 11-1: Infinite Sequences. While we are used to dealing with continuous functions in Calculus, sequences allow us to look at ordered lists of numbers. Understanding sequences is the foundational step before we tackle infinite series.

What is an Infinite Sequence?

An infinite sequence is essentially a list of numbers written in a definite order:

$$a_1, a_2, a_3, a_4, \dots, a_n, \dots$$

Here, $a_1$ is the first term, $a_2$ is the second, and $a_n$ represents the general $n$th term. We often denote a sequence as $\{a_n\}$. For example, if we have the sequence defined by $a_n = \frac{(-1)^n(n+1)}{3^n}$, we can find the first few terms by plugging in $n=1, 2, 3, \dots$ to generate the list.

Convergence vs. Divergence

The most important question we ask about a sequence is: does it have a limit?

  • Convergent: If the terms approach a specific finite number $L$ as $n$ goes to infinity (written as $\lim_{n \to \infty} a_n = L$), the sequence converges.
  • Divergent: If the limit does not exist (it oscillates without settling) or goes to $\pm \infty$, the sequence diverges.

Tools for Evaluating Limits

Finding the limit of a sequence is very similar to finding limits of functions from Calculus I. Here are the key techniques from our class notes:

1. L'Hopital's Rule

If you encounter an indeterminate form like $\frac{\infty}{\infty}$, you can treat the sequence like a function and apply L'Hopital's Rule.
Example: For $\lim_{n \to \infty} \frac{\ln n}{n}$, taking the derivative of the top and bottom gives us $\lim_{n \to \infty} \frac{1/n}{1} = 0$. Therefore, the sequence converges to 0.

2. The Absolute Value Theorem

A useful theorem states that if $\lim_{n \to \infty} |a_n| = 0$, then $\lim_{n \to \infty} a_n = 0$. This is particularly helpful for alternating sequences with terms like $(-1)^n$.

3. The Squeeze Theorem

If we can sandwich our sequence between two other sequences that converge to the same limit, our sequence must also converge to that limit.
Example: Consider $a_n = \frac{\cos^2 n}{2^n}$. Since $0 \le \cos^2 n \le 1$, we can write:

$$0 \le \frac{\cos^2 n}{2^n} \le \frac{1}{2^n}$$

Because the limit of $\frac{1}{2^n}$ is 0, by the Squeeze Theorem, our sequence also converges to 0.

4. Working with Factorials

Factorials ($n!$) grow very fast. When simplifying ratios of factorials, expand the larger factorial until it matches the smaller one to cancel terms out.
Example:

$$a_n = \frac{(2n-1)!}{(2n+1)!} = \frac{(2n-1)!}{(2n+1)(2n)(2n-1)!} = \frac{1}{(2n+1)(2n)}$$

As $n \to \infty$, the denominator grows infinitely large, so the limit is 0.

5. Logarithm Properties

Don't forget your log laws! For a sequence like $a_n = \ln(n+1) - \ln n$, you can combine the terms:

$$\lim_{n \to \infty} \ln\left(\frac{n+1}{n}\right) = \ln(1) = 0$$

Mastering these techniques will make determining the behavior of sequences much more intuitive. Keep practicing those algebraic simplifications, and I'll see you in the next class!