Chapter 11 Test Review - Ace Your Exam!

Welcome to the Chapter 11 Test Review for Professor Baker's Math Class! This review will help you solidify your understanding of series and sequences. We'll go over key concepts and provide examples to boost your confidence before the test.

Review Sessions

  • Review Part 1: April 18, 2024
  • Review Part 2: April 23, 2024

Key Concepts and Practice Problems

Here are some of the key concepts and practice problems you should be familiar with for the test:

1. Convergence and Divergence Tests

Determining whether a series converges or diverges is a fundamental skill. Here's a reminder of some common tests:

  • Test for Divergence: If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
  • p-Series: The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$.
  • Geometric Series: The series $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1-r}$ if $|r| < 1$ and diverges if $|r| \ge 1$.
  • Comparison Tests: Use direct comparison or limit comparison test with known series (p-series, geometric series).
  • Alternating Series Test: For an alternating series $\sum_{n=1}^{\infty} (-1)^{n-1}b_n$, if $b_n > 0$, $b_{n+1} \le b_n$ for all $n$, and $\lim_{n \to \infty} b_n = 0$, then the series converges.
  • Ratio Test: Useful for series involving factorials or exponential terms. Let $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. If $L < 1$, the series converges absolutely. If $L > 1$, the series diverges. If $L = 1$, the test is inconclusive.
  • Root Test: Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. If $L < 1$, the series converges absolutely. If $L > 1$, the series diverges. If $L = 1$, the test is inconclusive.
  • Integral Test: If $a_n = f(n)$ where $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$, then $\sum_{n=1}^{\infty} a_n$ and $\int_1^{\infty} f(x) dx$ either both converge or both diverge.

2. Radius and Interval of Convergence

To find the radius and interval of convergence for a power series $\sum_{n=0}^{\infty} c_n(x-a)^n$:

  1. Use the Ratio Test to find the values of $x$ for which the series converges.
  2. The radius of convergence, $R$, is related to the limit found in the Ratio Test.
  3. Check the endpoints of the interval to determine the interval of convergence.

3. Indefinite Integrals as Infinite Series

To evaluate an indefinite integral as an infinite series:

  1. Express the integrand as a power series. For example, recall that $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$. Therefore, $\sin(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n (x^2)^{2n+1}}{(2n+1)!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+2}}{(2n+1)!}$.
  2. Integrate the power series term by term.
  3. Add the constant of integration, $C$.

For example, consider $\int x^2 \sin(x^2) dx$. Using the power series for $\sin(x^2)$, we get $\int x^2 \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+2}}{(2n+1)!} dx = \int \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+4}}{(2n+1)!} dx = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+5}}{(2n+1)!(4n+5)} + C$.

Remember to practice, review your notes, and ask questions! You've got this!