Subsection A.7 Summary of Tests for Series
Test for Divergence.
\(\displaystyle {\sum ^\infty _{n=1}{a_n}}\)
Diverges if \(\displaystyle {\lim _{n \rightarrow \infty } a_n \ne 0}\)
Note: This test cannot be used to show convergence.
Geometric Series.
\(\displaystyle {\sum ^\infty _{n=0}{r^n}}\)
Converges if \(\left| r \right| \lt 1\)
Diverges if \(\left| r \right| \lt 1\)
Note: When convergent, \(\displaystyle {\text{Sum} = \frac{1}{1-r}}\)
Telescoping Series.
\(\displaystyle {\sum ^\infty _{n=1}{(b_n-b_{n+a})}}\)
Converges if \(\displaystyle {\lim _{n \rightarrow \infty } b_n = L}\)
Note: When convergent, \(\displaystyle \text{Sum}= \left(\sum ^a_{n=1}b_n\right) -L\)
\(p\)-Series.
\(\displaystyle {\sum ^\infty _{n=1}{\frac{1}{(an+b)^p}}}\)
Converges if \(p\gt1\)
Diverges if \(p\le 1\)
Integral Test.
\(\displaystyle {\sum ^\infty _{n=0}{a_n}}\)
Converges if \(\displaystyle \int _1^\infty a(n)\ dn\) is convergent
Diverges if \(\displaystyle \int _1^\infty a(n)\ dn\) is divergent
Note: \(a_n = a(n)\) must be continuous.
Direct Comparison.
\(\displaystyle {\sum ^\infty _{n=0}{a_n}}\)
Converges if \(\displaystyle \sum _{n=0}^\infty b_n \) converges and \(0\le a_n\le b_n\)
Diverges if \(\displaystyle \sum _{n=0}^\infty b_n \) diverges and \(0\le b_n\le a_n\)
Limit Comparison.
\(\displaystyle {\sum ^\infty _{n=0}{a_n}}\)
Converges if \(\displaystyle \sum _{n=0}^\infty b_n \) converges and \(\displaystyle \lim _{n\rightarrow \infty } a_n/b_n \ge 0\)
Diverges if \(\displaystyle \sum _{n=0}^\infty b_n \) diverges and \(\displaystyle \lim _{n\rightarrow \infty } a_n/b_n \gt 0\)
Ratio Test.
\(\displaystyle {\sum ^\infty _{n=0}{a_n}}\)
Converges if \(\displaystyle \lim _{n\rightarrow \infty } \frac{a_{n+1}}{a_n} \lt 1\)
Diverges if \(\displaystyle \lim _{n\rightarrow \infty } \frac{a_{n+1}}{a_n} \gt 1\)
Note: \(\lbrace a_n\rbrace \) must be positive. Also diverges if \(\displaystyle \lim _{n\rightarrow \infty } a_{n+1}/a_n=\infty \text{.}\)
Root Test.
\(\displaystyle {\sum ^\infty _{n=0}{a_n}}\)
Converges if \(\displaystyle \lim _{n\rightarrow \infty } \big (a_n\big )^{1/n} \lt 1\)
Diverges if \(\displaystyle \lim _{n\rightarrow \infty } \big (a_n\big )^{1/n} \gt 1\)
Note: \(\lbrace a_n\rbrace \) must be positive. Also diverges if \(\displaystyle \lim _{n\rightarrow \infty } \big (a_n\big )^{1/n}=\infty \text{.}\)