Good Evening,

This week we're covering some exciting convergence tests! Here's a breakdown of the key concepts from Sections 11.5 and 11.6. Remember, I'm available in my Online Office from 5 pm to 6 pm on March 31st if you need help with any topics in Chapter 11.

Materials for this week:

Video Resources:

Key Concepts:

1. Alternating Series Test:

An alternating series is a series whose terms alternate in sign. It can be written in the form:

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

The Alternating Series Test states that if the alternating series $ \sum_{n=1}^{\infty} (-1)^{n-1} b_n $ satisfies:

  1. $b_{n+1} \le b_n$ for all $n$ (i.e., the terms are decreasing)
  2. $\lim_{n \to \infty} b_n = 0$

Then the series is convergent.

Example: Consider the alternating harmonic series:

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

Here, $b_n = \frac{1}{n}$. Since $b_{n+1} = \frac{1}{n+1} \le \frac{1}{n} = b_n$ and $\lim_{n \to \infty} \frac{1}{n} = 0$, the alternating harmonic series converges by the Alternating Series Test.

2. Absolute Convergence:

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

Example: The series $ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2} = -1 + \frac{1}{4} - \frac{1}{9} + \dots $ is absolutely convergent because $ \sum_{n=1}^{\infty} \left| \frac{(-1)^{n-1}}{n^2} \right| = \sum_{n=1}^{\infty} \frac{1}{n^2} $ is a convergent p-series (p = 2).

3. Conditional Convergence:

A series $ \sum a_n $ is called conditionally convergent if it is convergent but not absolutely convergent.

Theorem: If a series $ \sum a_n $ is absolutely convergent, then it is convergent. This means absolute convergence implies convergence.

4. Ratio Test:

Let $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Then:

  • If $L < 1$, the series $ \sum a_n $ is absolutely convergent (and therefore convergent).
  • If $L > 1$ or $L = \infty$, the series $ \sum a_n $ is divergent.
  • If $L = 1$, the Ratio Test is inconclusive.

5. Root Test:

Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} |a_n|^{\frac{1}{n}}$. Then:

  • If $L < 1$, the series $ \sum a_n $ is absolutely convergent (and therefore convergent).
  • If $L > 1$ or $L = \infty$, the series $ \sum a_n $ is divergent.
  • If $L = 1$, the Root Test is inconclusive.

Remember to practice applying these tests to various series! Good luck, and see you in office hours!

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