test the convergence of sigma n! /3^n
Answers
Answer:
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,
[sum notation] and [more sum notation].
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.