Class Notes: April 4, 2023 - Sections 11.5 & 11.6

Welcome back to Professor Baker's Math Class! Today, we delved into the fascinating world of infinite series, focusing on how to determine whether they converge or diverge. We covered the Alternating Series Test, absolute convergence, the Ratio Test and the Root Test. Let's recap the key concepts:

Alternating Series Test (AST)

The Alternating Series Test is a powerful tool for determining the convergence of alternating series, which have terms that alternate in sign. The general form of an alternating series is:

$$ \sum_{n=1}^{\infty} (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + b_5 - b_6 + ... $$, where $b_n > 0$ for all $n$.

To apply the AST, the alternating series must satisfy these conditions:

  1. $b_{n+1} \le b_n$ for all $n$ (the terms are decreasing in magnitude).
  2. $\lim_{n \to \infty} b_n = 0$ (the terms approach zero).

If both conditions are met, then the alternating series converges!

Example: The alternating harmonic series: $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$

  • $b_n = \frac{1}{n}$, which is alternating.
  • $\lim_{n \to \infty} \frac{1}{n} = 0$.
  • $\frac{1}{n+1} \le \frac{1}{n}$ for all $n$.

Therefore, the alternating harmonic series converges by the AST!

Absolute Convergence

A series $\sum a_n$ is said to be absolutely convergent if the series of absolute values, $\sum |a_n|$, is convergent. If a series is absolutely convergent, then it is also convergent.

Example: Consider the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2}$. Taking the absolute value gives us $\sum_{n=1}^{\infty} \frac{1}{n^2}$, which is a p-series with $p = 2 > 1$. Thus, $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges. Therefore, $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2}$ is absolutely convergent.

The Ratio Test

The Ratio Test is useful for series involving factorials or exponential terms. Let $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. Then:

  • If $L < 1$, the series $\sum a_n$ converges absolutely.
  • If $L > 1$ or $L = \infty$, the series $\sum a_n$ diverges.
  • If $L = 1$, the Ratio Test is inconclusive.

The Root Test

The Root Test is another useful tool for determining convergence. Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. Then:

  • If $L < 1$, the series $\sum a_n$ converges absolutely.
  • If $L > 1$ or $L = \infty$, the series $\sum a_n$ diverges.
  • If $L = 1$, the Root Test is inconclusive.

Keep practicing, and you'll master these tests in no time! You've got this!