Chapter 11 Section 7: Strategy for Testing Series

Welcome to Section 7 of Chapter 11! Tonight, we're diving into the exciting world of infinite sequences and series, focusing on how to determine whether a series converges or diverges. This is a crucial skill in calculus, so let's get started!

Here are the resources for tonight's lesson:

Key Concepts and Strategies

Choosing the right test for convergence or divergence can seem tricky, but with practice, you'll become a pro! Here's a breakdown of strategies and tests to consider:

  1. P-Series: If your 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: For a series like $\sum ar^{n-1}$ or $\sum ar^n$, it's geometric. It converges if $|r| < 1$ and diverges if $|r| \ge 1$. Algebraic manipulation might be needed to get it in this form.
  3. Comparison Tests: If your series resembles a p-series or a geometric series, consider using the Comparison Test or the Limit Comparison Test. These are particularly useful when $a_n$ is a rational or algebraic function of $n$. Remember, these tests apply to series with positive terms. If there are negative terms, consider testing for absolute convergence by applying the comparison test to $\sum |a_n|$.
  4. Test for Divergence: Always a good first check! If $\lim_{n \to \infty} a_n \neq 0$, then the series diverges. This one can save you a lot of time.
  5. Alternating Series Test: If your series is in the form $\sum (-1)^{n-1}b_n$ or $\sum (-1)^n b_n$, the Alternating Series Test is a strong candidate. You need to verify that $b_n > 0$, $b_{n+1} \le b_n$ for all $n$, and $\lim_{n \to \infty} b_n = 0$.
  6. Ratio Test: This test is excellent for series involving factorials or other products (including constants raised to the $n$th power). Keep in mind that $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} \to 1$ for p-series and rational algebraic functions, so avoid using the Ratio Test for those.
  7. Root Test: If $a_n$ is of the form $(b_n)^n$, give the Root Test a try.
  8. Integral Test: If $a_n = f(n)$ and $\int_1^{\infty} f(x) dx$ is easily evaluated, the Integral Test could be effective (assuming the hypotheses are met).

Examples

Let's consider a few examples to illustrate these strategies:

  • Example 1: $\sum_{n=1}^{\infty} \frac{2n+1}{n}$. Since $\lim_{n \to \infty} \frac{2n+1}{n} = 2 \neq 0$, we should use the Test for Divergence.
  • Example 2: $\sum_{n=1}^{\infty} \frac{\sqrt{n^3 + 1}}{3n^3 + 4n^2 + 2}$. Since $a_n$ is an algebraic function of $n$, we can compare this with a p-series using the Limit Comparison Test.
  • Example 4: $\sum_{n=1}^{\infty} (-1)^n \frac{n^3}{n^4 + 1}$. Since the series is alternating, we use the Alternating Series Test.
  • Example 5: $\sum_{k=1}^{\infty} \frac{2^k}{k!}$. Since the series involves $k!$, we use the Ratio Test.

Remember, practice is key! Work through various problems, and you'll develop an intuition for which test to apply. Don't be afraid to experiment and learn from your mistakes. You've got this!