Section B.7 Summary of Tests for Series
| 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 |
\(\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n}=\infty\) |
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| 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 |
\(\lim\limits_{n\to\infty} (a_n)^{1/n}=\infty\) |