# List of Series Tests

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,

and .

Click on the name of the test to get more information on the test.

## 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.

Please note that this does not mean that the sum of the series is that same as the value of the integral. In most cases, the two will be quite different.

### 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.