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APEX Calculus: for University of Lethbridge

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