AP Calculus BC / Unit 10: Infinite Sequences and Series
10.8 Ratio Test for Convergence
Notes
Let $\displaystyle L=\lim\left|\frac{a_{n+1}}{a_n}\right|$.
- $L<1$: converges absolutely. $L>1$: diverges. $L=1$: inconclusive.
- Great for factorials and exponentials.
Practice
For $\displaystyle \sum\frac{1}{n!}$, the ratio limit is:
$\displaystyle \frac{1}{n+1}\to0$, so it converges.
For $\displaystyle \sum\frac{3^n}{n}$, the ratio limit is:
$\displaystyle \frac{3n}{n+1}\to3>1$, so it diverges.
The ratio test is inconclusive for $\displaystyle \sum\frac{1}{n^2}$ because:
Use the p-series test instead.
Ready for more? Take the Unit 10 test.