Welcome back, class! We have arrived at one of the most crucial moments in Calculus II: Chapter 11 on Infinite Series. This chapter often challenges students because it requires not just calculation, but strategy. You need to look at a summation and instantly recognize which tool from your mathematical toolkit is the best fit for the job.
What's in this Post?
Attached below, you will find two important documents:
- The Chapter 11 Take-Home Test: A set of 10 problems where you must determine convergence or divergence.
- Class Notes & Strategy Guide: A breakdown of the convergence tests, including the Ratio Test, Root Test, and Comparison Tests, along with solved examples.
The Strategy for Testing Convergence
As you review the attached notes, keep this general strategy in mind when approaching the test problems:
- Test for Divergence: Always check $\lim_{n \to \infty} a_n$ first. If the limit is not 0, the series diverges immediately. (See Problem #41 in the notes).
- p-Series & Geometric Series: Is the series of the form $\sum \frac{1}{n^p}$ or $\sum ar^{n-1}$? These are the easiest to spot. Remember, geometric series converge only if $|r| < 1$.
- Ratio Test: This is your best friend for factorials ($n!$) and exponentials. For example, looking at Problem 10 on your test ($\sum_{n=1}^{\infty} \frac{3^{n+1}}{(n+1)!}$), the presence of the factorial is a huge hint to use the Ratio Test: $$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $$
- Root Test: If you see expressions raised to the $n$-th power, like in Problem 2 ($\sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}}$), the Root Test will likely simplify the algebra significantly.
- Comparison Tests (Direct & Limit): For rational functions or algebraic functions involving roots (like Problem 1 and Problem 9), compare the series to a standard p-series that behaves similarly.
- Alternating Series Test (AST): If you see terms like $(-1)^n$ or $(-1)^{n-1}$, such as in Problem 6, verify that the terms are decreasing and the limit approaches zero.
Tips for the Test
The instructions on the test are specific: "All answers must include the test used and any work to show that the test works."
Simply writing "Converges" is not enough. You must prove it. For instance, if you use the Integral Test (a strong candidate for Problem 3: $\sum_{n=2}^{\infty} \frac{1}{n\sqrt{\ln n}}$), you must show that the function is continuous, positive, and decreasing on the interval.
Take your time with the algebraic manipulation, especially with exponents and factorials. Good luck, and happy studying!