AP Calculus BC / Unit 10: Infinite Sequences and Series
10.9 Determining Absolute or Conditional Convergence
Notes
- Absolutely convergent: $\displaystyle \sum|a_n|$ converges.
- Conditionally convergent: $\displaystyle \sum a_n$ converges but $\displaystyle \sum|a_n|$ diverges.
- Absolute convergence implies convergence.
Practice
$\displaystyle \sum\frac{(-1)^n}{n}$ is:
It converges, but $\displaystyle \sum\frac{1}{n}$ diverges.
$\displaystyle \sum\frac{(-1)^n}{n^2}$ is:
$\displaystyle \sum\frac{1}{n^2}$ converges.
If $\displaystyle \sum|a_n|$ converges, then $\displaystyle \sum a_n$:
Absolute convergence implies convergence.
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