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Section B.7 Summary of Tests for Series
Table B.7.1.
| Test |
Series |
Condition(s) of Convergence |
Condition(s) of Divergence |
Comment |
|
\(n\)th-Term |
\(\displaystyle{\sum^\infty_{n=1}{a_n}}\) |
|
\(\displaystyle{\lim_{n \to \infty} a_n \neq 0}\) |
Cannot be used to show convergence. |
| Geometric Series |
\(\displaystyle{\sum^\infty_{n=0}{r^n}}\) |
\(\abs{r} \lt 1\) |
\(\abs{r} \geq 1\) |
\(\displaystyle{\text{ Sum } = \frac{1}{1-r}}\) |
| Telescoping Series |
\(\displaystyle{\sum^\infty_{n=1}{(b_n-b_{n+a})}}\) |
\(\displaystyle{\lim_{n \to \infty} b_n = L}\) |
|
\(\displaystyle\text{ Sum } = \left(\sum^a_{n=1}b_n\right) -L\) |
|
\(p\)-Series |
\(\displaystyle{\sum^\infty_{n=1}{\frac{1}{(an+b)^p}}}\) |
\(p \gt 1\) |
\(p\leq 1\) |
|
| Integral Test |
\(\displaystyle{\sum^\infty_{n=0}{a_n}}\) |
\(\displaystyle \int_1^\infty a(n)\, dn\) converges |
\(\displaystyle \int_1^\infty a(n)\, dn\) diverges |
\(a_n = a(n)\) must be continuous |
| Direct Comparison |
\(\displaystyle{\sum^\infty_{n=0}{a_n}}\) |
\(\displaystyle \sum_{n=0}^\infty b_n\) converges and \(0\leq a_n\leq b_n\)
|
\(\displaystyle \sum_{n=0}^\infty b_n\) diverges and \(0\leq b_n\leq a_n\)
|
|
| Limit Comparison |
\(\displaystyle{\sum^\infty_{n=0}{a_n}}\) |
\(\displaystyle \sum_{n=0}^\infty b_n\) converges and \(\lim\limits_{n\to\infty}\frac{a_n}{b_n} \geq 0\)
|
\(\displaystyle \sum_{n=0}^\infty b_n\) diverges and \(\lim\limits_{n\to\infty}\frac{a_n}{b_n} \gt 0\)
|
Also diverges if \(\lim\limits_{n\to\infty}\frac{a_n}{b_n}=\infty\)
|
| Ratio Test |
\(\displaystyle{\sum^\infty_{n=0}{a_n}}\) |
\(\displaystyle \lim_{n\to\infty} \frac{a_{n+1}}{a_n} \lt 1\) |
\(\displaystyle \lim_{n\to\infty} \frac{a_{n+1}}{a_n} \gt 1\) |
\(\{a_n\}\) must be positive |
|
|
|
Also diverges if |
\(\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n}=\infty\) |
| Root Test |
\(\displaystyle{\sum^\infty_{n=0}{a_n}}\) |
\(\displaystyle \lim_{n\to\infty} \big(a_n\big)^{1/n} \lt 1\) |
\(\displaystyle \lim_{n\to\infty} \big(a_n\big)^{1/n} \gt 1\) |
\(\{a_n\}\) must be positive |
|
|
|
Also diverges if |
\(\lim\limits_{n\to\infty} (a_n)^{1/n}=\infty\) |
