Determine whether the series is absolutely convergent condit

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. You must justify your conclusion, including any tests used and evidence that the requirements of the tests are met. sigma_n = 1^infinity (-1)^n In n/n sigma_n = 1^infinity 4^n/n^2n!

Solution

1) We will use Alternating series test .

(ln n )/n goes to 0 as n goes to infinty and also ln/n is a decreasing sequence.

hence it satisfies all conditions in Alternative series test. And hence it is absolutely convergent.

2)4^n/(n^2 n!) is less than 1/n^2 for large n . hence it is convergent

this test is the sandwich test

series Sum(1/n^2) is convergent