Chapter 11-5: Alternating Series

In this section, we'll learn about alternating series, where the terms alternate in sign. These series have the form:

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

To determine if an alternating series converges, we can use the Alternating Series Test. This test has two 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 is convergent. A classic example is the alternating harmonic series:

$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ... = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$

This series converges by the Alternating Series Test because $\frac{1}{n+1} < \frac{1}{n}$ and $\lim_{n \to \infty} \frac{1}{n} = 0$.

Also important is the Alternating Series Estimation Theorem, which states that if a series satisfies the conditions of the Alternating Series Test, then the error in approximating the sum of the series by its $n$th partial sum $s_n$ is no greater than the absolute value of the $(n+1)$th term. In other words, $|R_n| = |s - s_n| \le b_{n+1}$.

For example, to find the sum of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}$ correct to three decimal places, we need to find $n$ such that $\frac{1}{(n+1)!} < 0.0002$.

Chapter 11-6: Absolute Convergence and the Ratio and Root Tests

A series $\sum a_n$ is absolutely convergent if the series of absolute values $\sum |a_n|$ is convergent.

Important Theorem: If a series is absolutely convergent, then it is convergent.

A series that is convergent but not absolutely convergent is called conditionally convergent. The alternating harmonic series is an example of conditional convergence.

To determine if a series is absolutely convergent, we can use the Ratio Test:

  1. If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L < 1$, then the series is absolutely convergent.
  2. If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L > 1$ or $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = \infty$, then the series is divergent.
  3. If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = 1$, the Ratio Test is inconclusive.

Another test for absolute convergence is the Root Test:

  1. If $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L < 1$, then the series is absolutely convergent.
  2. If $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L > 1$ or $\lim_{n \to \infty} \sqrt[n]{|a_n|} = \infty$, then the series is divergent.
  3. If $\lim_{n \to \infty} \sqrt[n]{|a_n|} = 1$, the Root Test is inconclusive.

Keep practicing, and you'll master these concepts in no time! Good luck!