| Line 5: | Line 5: | ||
1. The nth-Term Test: | 1. The nth-Term Test: | ||
| − | Unless <math> | + | Unless <math>a_{n}\rightarrow 0</math>, the series diverges |
2. Geometric series: | 2. Geometric series: | ||
| Line 17: | Line 17: | ||
4. Ratio Test(for series with non-negative terms): | 4. Ratio Test(for series with non-negative terms): | ||
| − | <math> \lim \limits_{n \to \infty }{\frac{ | + | <math> \lim \limits_{n \to \infty }{\frac{a_{n+1}}{a_{n}}}=p</math> 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): | 5. Root Test(for series with non-negative terms): | ||
| − | <math> \lim \limits_{n \to \infty }{\sqrt[n]{ | + | <math> \lim \limits_{n \to \infty }{\sqrt[n]{a_{n}} }=p</math> The series converges if p<1, diverges if p>1, and is inconclusive if p=1 |
More to come... | More to come... | ||
Revision as of 16:33, 9 November 2008
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 $ a_{n}\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{a_{n+1}}{a_{n}}}=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]{a_{n}} }=p $ The series converges if p<1, diverges if p>1, and is inconclusive if p=1
More to come...
