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Handouts on the comparison tests for determining the convergence or divergence of series. The tests include the direct comparison test and the limit comparison test. The direct comparison test states that if the terms of series 'a' are less than or equal to the terms of series 'b' for all indices greater than a certain value 'n0' and series 'b' converges, then series 'a' also converges. Conversely, if the terms of series 'a' are greater than or equal to the terms of series 'b' for all indices greater than 'n0' and series 'b' diverges, then series 'a' also diverges. The limit comparison test states that if the limit of the ratio of the terms of series 'a' to series 'b' as n approaches infinity is a finite and positive number, then both series converge or both series diverge.
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Handouts for Section 12.4: The comparison tests
an and
bn are series with positive terms.
(a) If an ≤ bn for all n ≥ n 0 and
bn converges, then
an converges. (b) If an ≥ bn for all n ≥ n 0 and
bn diverges, then
an diverges.
an and
bn are series with positive terms. If
nlim→∞
an bn
= c
where c is a finite number and c > 0, then either both series converge or both series diverge.
