I saw a really great study guide that got passed around before the first Engineering 195 test, so I thought it would be a great idea to create a collaborative study guide for MA 181.
Convergent and Divergent Series Tests
1. The nth-Term Test:
Unless $ an\rightarrow 0 $, the series diverges
2. Geometric series:
$ \sum_{n=1}^\infty ar^n $ converges if $ |r| < 1 $; otherwise it diverges.
3. p-series:
$ \sum_{n=1}^\infty 1/n^p $ converges if $ p>1 $; otherwise it diverges
4. Ratio Test(for series with non-negative terms):
$ \lim \limits_{n \to \infty }{\frac{an+1}{an}}=p $ The series converges if p<1, diverges if p>1, and is inconclusive if p=1
5. Root Test(for series with non-negative terms):
$ \lim \limits_{n \to \infty }{\sqrt[n]{an} }=p $ The series converges if p<1, diverges if p>1, and is inconclusive if p=1
