Welcome to Chapter 11-4 and 11-5!

Hello Math Students! Today, we're diving into sections 11-4 and 11-5 of our textbook. These sections cover some powerful techniques for analyzing infinite series. We'll be focusing on the Comparison Tests and Alternating Series Tests.

Section 11-4: The Comparison Tests

The Comparison Tests help us determine whether a series converges or diverges by comparing it to a series whose behavior we already know. The core idea is beautifully simple: if a series is smaller than a known convergent series, it also converges; if it's larger than a known divergent series, it diverges as well! Let's break this down:

  • Direct Comparison Test: Suppose we have two series, $\sum a_n$ and $\sum b_n$, with positive terms.
    • If $\sum b_n$ converges and $a_n \le b_n$ for all $n$, then $\sum a_n$ also converges.
    • If $\sum b_n$ diverges and $a_n \ge b_n$ for all $n$, then $\sum a_n$ also diverges.

For example, consider the series $\sum_{n=1}^{\infty} \frac{1}{2^n + 1}$. This series *feels* convergent because it's similar to the geometric series $\sum_{n=1}^{\infty} \frac{1}{2^n}$, which we know converges. And indeed, since $\frac{1}{2^n + 1} < \frac{1}{2^n}$, the Direct Comparison Test confirms our intuition!

Sometimes, direct comparison isn't straightforward. That's where the Limit Comparison Test comes in!

  • Limit Comparison Test: Suppose we have two series, $\sum a_n$ and $\sum b_n$, with positive terms. If $$\lim_{n \to \infty} \frac{a_n}{b_n} = c$$ where $c$ is a finite number and $c > 0$, then either both series converge or both series diverge.

This test allows you to compare series even when the inequality $a_n \le b_n$ or $a_n \ge b_n$ doesn't hold directly. The limit comparison test basically says that if the two series are similar as $n$ becomes very large, then they both either converge or diverge.

Key Series for Comparison:

  • p-series: The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$.
  • Geometric series: The series $\sum_{n=1}^{\infty} ar^{n-1}$ converges if $|r| < 1$ and diverges if $|r| \ge 1$.

Section 11-5: Alternating Series

An alternating series is a series where the terms alternate in sign. They have a general form like:

$$\sum_{n=1}^{\infty} (-1)^{n-1}b_n = b_1 - b_2 + b_3 - b_4 + ...$$

Where $b_n > 0$ for all $n$

  • Alternating Series Test (AST): If the alternating series satisfies the following two conditions then the series converges:
    1. $b_{n+1} \le b_n$ for all $n$ (i.e., the terms are decreasing in magnitude)
    2. $\lim_{n \to \infty} b_n = 0$

Example: The alternating harmonic series: $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ...$. Applying the AST, we see that: 1. $\frac{1}{n+1} \le \frac{1}{n}$ for all $n$ so the terms are decreasing 2. $\lim_{n \to \infty} \frac{1}{n} = 0$. Thus the alternating harmonic series converges.

Keep practicing, and you'll master these convergence tests in no time. Good luck, and happy calculating!