Good Morning, Math Students!

This week we're focusing on Section 11-7, a crucial section for understanding and reviewing series. I had some technical difficulties with the videos for Sections 11-8 and 11-9, but I'm re-recording them, so stay tuned! In the meantime, dedicating your time to mastering Section 11-7 will be highly beneficial. It serves as excellent review since it requires you to synthesize all the convergence/divergence tests we've learned.

Focus on Section 11-7: Strategy for Testing Series

Section 11-7 is all about developing a strategy for testing series. You'll be presented with a series and need to decide which test to apply. It’s similar to integrating functions - there isn't always one "right" way, but some approaches are more efficient than others. Remember, don't just apply the tests in order! Classify the series according to its form to determine the best approach.

Here's a breakdown of key strategies from the notes:

  1. p-series: If the series is in the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, it's a p-series. Remember: it converges if $p > 1$ and diverges if $p \le 1$.
  2. Geometric Series: If the series is in the form $\sum ar^{n-1}$ or $\sum ar^n$, it's a geometric series. It converges if $|r| < 1$ and diverges if $|r| \ge 1$. Sometimes, algebraic manipulation is needed to get it into this form.
  3. Comparison Tests: If the series looks similar to a p-series or a geometric series, consider the comparison tests (Direct Comparison Test or Limit Comparison Test). These are particularly useful when $a_n$ is a rational or algebraic function of $n$ (involving roots of polynomials). Remember the comparison tests apply to series with positive terms. If $a_n$ has negative terms consider using the comparison tests on $\sum |a_n|$ to test for absolute convergence.
  4. Test for Divergence: If you can easily see that $\lim_{n \to \infty} a_n \ne 0$, then the Test for Divergence should be your first choice!
  5. Alternating Series Test: If the series is in the form $\sum (-1)^{n-1}b_n$ or $\sum (-1)^n b_n$, where $b_n > 0$, the Alternating Series Test is likely the best option.
  6. Ratio Test: Series involving factorials or other products (including constants raised to the $n$th power) are often easily tested using the Ratio Test. Keep in mind that $\frac{n+1}{n} \to 1$ as $n \to \infty$ for all p-series and rational/algebraic functions of $n$. Therefore, the Ratio Test is generally not helpful for p-series.
  7. Root Test: If $a_n$ is of the form $(b_n)^n$, the Root Test might be useful.
  8. Integral Test: If $a_n = f(n)$, and $\int_1^{\infty} f(x) dx$ is easily evaluated, the Integral Test can be effective (assuming $f(x)$ is continuous, positive, and decreasing).

Examples

Let's look at some examples from the notes:

  • $\sum_{n=1}^{\infty} \frac{2n+1}{n-1}$: Use the Test for Divergence, since $\lim_{n \to \infty} \frac{2n+1}{n-1} = 2 \neq 0$
  • $\sum_{n=1}^{\infty} \frac{\sqrt{n^3 + 1}}{3n^3 + 4n^2 + 2}$: Use the Limit Comparison Test with the p-series $\sum \frac{\sqrt{n^3}}{3n^3} = \sum \frac{n^{3/2}}{3n^3} = \sum \frac{1}{3n^{3/2}}$, since $\frac{\sqrt{n^3 + 1}}{3n^3 + 4n^2 + 2}$ is an algebraic function of $n$
  • $\sum_{n=1}^{\infty} (-1)^n \frac{n^3}{n^4 + 1}$: Use the Alternating Series Test, since the series is alternating.
  • $\sum_{k=1}^{\infty} \frac{2^k}{k!}$: Use the Ratio Test, since the series involves $k!$.
  • $\sum_{n=1}^{\infty} \frac{1}{2^n + 3^n}$: Use the Comparison Test with the geometric series $\sum \frac{1}{3^n}$, since the series is closely related to it.

Action Items:

  • Review the video and these notes carefully.
  • Work through the examples in the textbook.
  • Most importantly, practice, practice, practice! The more series you test, the better you'll become at identifying the best strategy.

Remember, I am here to support you. Please don't hesitate to ask questions in class or during office hours. Good luck, and happy studying!

Best,

Professor Baker