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List of Series Tests :

Return to the Series, Convergence, and Series Tests starting page
The series of interest will always by symbolized as the sum, as n goes from 1 to infinity, of a[n]. In addition, any auxilliary sequence will be symbolized as the sum, as n goes from 1 to infinity, of b[n]. Or, symbolically,

List of Series Tests :

Return to the Series, Convergence, and Series Tests starting page
The series of interest will always by symbolized as the sum, as n goes from 1 to infinity, of a[n]. In addition, any auxilliary sequence will be symbolized as the sum, as n goes from 1 to infinity, of b[n]. Or, symbolically,

The Common Series Tests

Divergence Test :

If the limit of a[n] is not zero, or does not exist, then the sum diverges.

Integral Test :

If you can define f so that it is a continuous, positive, decreasing function from 1 to infinity (including 1) such that a[n]=f(n), then the sum will converge if and only if the integral of f from 1 to infinity converges.

Comparison Test :

Let b[n] be a second series. Require that all a[n] and b[n] are positive. If b[n] converges, and a[n]<=b[n] for all n, then a[n] also converges. If the sum of b[n] diverges, and a[n]>=b[n] for all n, then the sum of a[n] also diverges.

Limit Comparison Test :

Let b[n] be a second series. Require that all a[n] and b[n] are positive.

If the limit of a[n]/b[n] is positive, then the sum of a[n] converges if and only if the sum of b[n] converges.
If the limit of a[n]/b[n] is zero, and the sum of b[n] converges, then the sum of a[n] also converges.
If the limit of a[n]/b[n] is infinite, and the sum of b[n] diverges, then the sum of a[n] also diverges.

Alternating Series Test :

If a[n]=(-1)^(n+1)b[n], where b[n] is positive, decreasing, and converging to zero, then the sum of a[n] converges.

Absolute Convergence Test :

If the sum of |a[n]| converges, then the sum of a[n] converges.

Ratio Test :

If the limit of |a[n+1]/a[n]| is less than 1, then the series (absolutely) converges. If the limit is larger than one, or infinite, then the series diverges.

Root Test :

If the limit of |a[n]|^(1/n) is less than one, then the series (absolutely) converges. If the limit is larger than one, or infinite, then the series diverges. 