AP Calculus BC / Unit 10: Infinite Sequences and Series
10.6 Comparison Tests for Convergence
Notes
For positive terms:
- Direct comparison: if $a_n\le b_n$ and $\displaystyle \sum b_n$ converges, $\displaystyle \sum a_n$ converges. If $a_n\ge b_n$ and $\displaystyle \sum b_n$ diverges, $\displaystyle \sum a_n$ diverges.
- Limit comparison: if $\displaystyle \lim\frac{a_n}{b_n}$ is a positive finite number, both do the same thing.
Practice
$\displaystyle \sum\frac{1}{n^2+1}$ converges by comparison with:
$\displaystyle \frac{1}{n^2+1}<\frac{1}{n^2}$.
$\displaystyle \sum\frac{n}{n^2+3}$, compared by limit with $\displaystyle \sum\frac{1}{n}$:
The ratio approaches 1, and the harmonic series diverges.
If $a_n\le b_n$ and $\displaystyle \sum b_n$ diverges, then $\displaystyle \sum a_n$:
Being smaller than a divergent series proves nothing.
Ready for more? Take the Unit 10 test.