We are trying something a bit different today as we navigate the transition to online learning. I understand that this is a unique and challenging time for everyone, and while the shift to digital classes wasn't our original plan, I am confident that you can be successful in this format.

To support your learning, I have uploaded the PowerPoint slides for today's lesson covering Chapter 11-3 and 11-4. Additionally, a full video lecture will be posted by 4:00 PM tonight.

Class Logistics & Support

  • Video Lecture: Available by 4pm.
  • Office Hours: I will be live on my WebEx channel today, March 26, from 5pm to 6pm.

Please utilize these office hours! If you have questions, issues accessing the material, or just need clarification on the math, pop in and ask. It is crucial that you view the lesson and keep up with the work to stay on track.

Lesson Overview: Infinite Sequences and Series

Today we are looking at powerful methods for determining if a series converges or diverges.

11.3 The Integral Test and Estimates of Sums

The Integral Test allows us to relate the convergence of a series to the convergence of an improper integral. If $f$ is a continuous, positive, and decreasing function on $[1, \infty)$ and $a_n = f(n)$, then:

$$ \int_1^{\infty} f(x) \, dx \text{ is convergent} \iff \sum_{n=1}^{\infty} a_n \text{ is convergent} $$

Key Concept: The p-series
A vital application of this is the $p$-series, $\sum_{n=1}^{\infty} \frac{1}{n^p}$. Remember this rule:

  • The series converges if $p > 1$.
  • The series diverges if $p \le 1$.

11.4 The Comparison Tests

Sometimes integration is too difficult. In these cases, we compare a complicated series to a known series (like a geometric series or a $p$-series).

Direct Comparison Test:
Suppose $\sum b_n$ is a series with positive terms.

  • If $\sum b_n$ is convergent and $a_n \le b_n$, then $\sum a_n$ is also convergent (smaller than a finite sum is finite).
  • If $\sum b_n$ is divergent and $a_n \ge b_n$, then $\sum a_n$ is also divergent (larger than infinity is infinity).

Limit Comparison Test:
If the Direct Comparison Test is inconclusive, we take the limit of the ratio of terms:

$$ \lim_{n \to \infty} \frac{a_n}{b_n} = c $$

If $c$ is a finite number and $c > 0$, then both series either converge or diverge together.

Please review the attached PDFs and the video lecture. If you need any help, please let me know or see me during office hours tonight!