Welcome back to Professor Baker's Math Class! Tonight, we are expanding our toolkit for analyzing infinite series. While we have previously looked at series with positive terms, we are now moving into Section 11.5 and Section 11.6 to handle series where terms may be negative or involve complex powers and factorials.

1. Alternating Series (Section 11.5)

An alternating series is one where the terms switch back and forth between positive and negative, usually looking like this:

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

To test if these converge, we use the Alternating Series Test. The series converges if two conditions are met:

  1. The terms are decreasing in magnitude: $b_{n+1} \le b_n$ for all $n$.
  2. The limit of the terms approaches zero: $\lim_{n \to \infty} b_n = 0$.

2. Absolute Convergence (Section 11.6)

We also explored the concept of Absolute Convergence. A series $\sum a_n$ is called absolutely convergent if the series of absolute values $\sum |a_n|$ converges.

  • Theorem: If a series is absolutely convergent, it is convergent.
  • Conditional Convergence: If $\sum a_n$ converges but $\sum |a_n|$ diverges, the series is conditionally convergent (e.g., the alternating harmonic series).

3. The Ratio Test

This is one of the most powerful tests in calculus, especially useful when dealing with factorials ($n!$) or exponentials. We calculate the limit of the ratio of consecutive terms:

$$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $$
  • If $L < 1$, the series is absolutely convergent.
  • If $L > 1$ (or $\infty$), the series is divergent.
  • If $L = 1$, the test is inconclusive (you must try a different test).

4. The Root Test

Similar to the Ratio Test, the Root Test is convenient when terms involve $n$-th powers. We look at the limit of the $n$-th root:

$$ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} $$

The conclusions are exactly the same as the Ratio Test: $L < 1$ implies convergence, while $L > 1$ implies divergence.

Please download the class notes and slides below to review the specific examples we worked through in class, including the alternating harmonic series and $p$-series comparisons.

Class Materials:
PPT | PDF | Zoom Meeting Link | Class Notes