Summary of Comparison Tests

Essential Concepts

  • The comparison tests are used to determine convergence or divergence of series with positive terms.
  • When using the comparison tests, a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is often compared to a geometric or p-series.

Glossary

comparison test
if [latex]0\le {a}_{n}\le {b}_{n}[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges; if [latex]{a}_{n}\ge {b}_{n}\ge 0[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges
limit comparison test
suppose [latex]{a}_{n},{b}_{n}\ge 0[/latex] for all [latex]n\ge 1[/latex]. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to L\ne 0[/latex], then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] both converge or both diverge; if [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to 0[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to \infty [/latex], and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges