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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

\(\lim\limits_{n\to\infty} (a_n)^{1/n}=\infty\)

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