Chapter 11 Test Review

Get ready for your Chapter 11 test! This review focuses on the key concepts of convergence and divergence of series, along with the application of power series. Remember to practice and don't hesitate to ask questions!

Key Topics

  • Finding Convergence and Divergence of Series: Learn different tests to determine if a series converges or diverges.
  • Using Power Series: Explore how to represent functions as power series and their applications.

Convergence and Divergence Tests

Here's a summary of tests that can be used to determine if a series converges or diverges. Remember to choose the appropriate test based on the form of the series.

  • Test for Divergence: If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges. It's important to note that if $\lim_{n \to \infty} a_n = 0$, the test is inconclusive, and another test is needed.
  • Comparison Test: Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms.
    • (i) If $\sum b_n$ is convergent and $a_n \le b_n$ for all $n$, then $\sum a_n$ is also convergent.
    • (ii) If $\sum b_n$ is divergent and $a_n \ge b_n$ for all $n$, then $\sum a_n$ is also divergent.
  • Limit Comparison Test: Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms. If $$\lim_{n \to \infty} \frac{a_n}{b_n} = c$$ where $c$ is a finite number and $c > 0$, then either both series converge or both diverge.
  • Integral Test: If $f$ is a continuous, positive, decreasing function on $[1, \infty)$ and let $a_n = f(n)$. Then the series $\sum_{n=1}^{\infty} a_n$ is convergent if and only if the improper integral $\int_{1}^{\infty} f(x) dx$ is convergent.
  • Alternating Series Test (AST): Consider an alternating series of the form $\sum_{n=1}^{\infty} (-1)^n b_n$ or $\sum_{n=1}^{\infty} (-1)^{n+1} b_n$, where $b_n > 0$ for all $n$. If
    • (i) $b_{n+1} \le b_n$ for all $n$ (i.e., the sequence {$b_n$} is decreasing), and
    • (ii) $\lim_{n \to \infty} b_n = 0$,
    then the alternating series converges.
  • Ratio Test: Let $\sum a_n$ be a series. Compute $$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$
    • If $L < 1$, then the series converges absolutely.
    • If $L > 1$ (or $L = \infty$), then the series diverges.
    • If $L = 1$, the Ratio Test is inconclusive.
  • Root Test: Let $\sum a_n$ be a series. Compute $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$
    • If $L < 1$, then the series converges absolutely.
    • If $L > 1$ (or $L = \infty$), then the series diverges.
    • If $L = 1$, the Root Test is inconclusive.

Examples

Refer to the attached notes for detailed examples demonstrating the application of these convergence and divergence tests.

Power Series

Power series are series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$, where $c_n$ are constants, $x$ is a variable, and $a$ is the center of the series.

  • Radius and Interval of Convergence: A power series converges for $|x - a| < R$ and diverges for $|x - a| > R$, where $R$ is the radius of convergence. The interval of convergence is the interval of $x$ values for which the series converges, and includes checking the endpoints $x = a \pm R$.

Good luck with your test! Remember to review your notes and practice problems.