Test | Series | Convergence or Divergence | Comments |
Ratio | ![]() |
If ![]()
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Inconclusive otherwise.
Useful for factorials and nth powers. For power series, let an be the entire power series term. |
nth-term | ![]() |
Diverges if ![]() |
Inconclusive if the limit is zero. |
geometric series | ![]() |
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Useful for comparison tests |
p-series | ![]() |
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Useful for comparison tests |
Integral | ![]() f(n) = an, f(x) defined for x>1 is continuous, positive, and decreasing (check f'(x) ) |
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Basic Comparison | ![]() usually ![]() all terms in both series strictly positive |
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Inconclusive otherwise.
Often the b-series is geometric or a p-series. Consider terms of the a-series of greatest effect. |
Limit Comparison | ![]() usually ![]() all terms in both series strictly positive |
If ![]() |
Inconclusive otherwise.
Often the b-series is geometric or a p-series. Consider terms of the a-series of greatest effect. |
Root | ![]() |
If ![]()
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Inconclusive otherwise.
Useful for nth powers, such as in power series. |
Absolute series | ![]() |
If ![]() ![]() |
Inconclusive otherwise.
Useful for series with both positive and negative terms. |
Alternating series | ![]() |
Converges if ![]() ![]() |
Inconclusive otherwise.
Note that an is the non-alternating part! If inequalities are not easy, let f(n) = an. f(x) defined for x>1 should be continuous, positive, and decreasing (check f'(x) ) |
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