Tests for Convergence and Divergence

Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms. 

nth-Term Test. A series is divergent if the nth term of the series does not approach zero as n becomes increasingly large. 

Ratio Test. If the absolute ratio of the (n + 1) term divided by the nth term as n becomes unbounded approaches 

Alternating-Series Leibniz Test. If the terms of a series are alternately positive and negative and never increase in value, the absolute series will converge, provided that the terms tend to zero as a limit. 

Cauchy s Root Test. If the nth root of the absolute value of the nth term, as n becomes unbounded, approaches 

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The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection Tests for Convergence and Divergence. ... 

Partial Sums of InBriite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent... 

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