Determine if is absolutely convergent, conditionally convergent or divergent.
A) absolutely convergent
B) conditionally convergent
C) divergent

Question

Determine if  is absolutely convergent, conditionally convergent or divergent.
A) absolutely convergent 
B) conditionally convergent 
C) divergent

Quizplus · Accepted Answer

The terms in the series contain alternating signs, so it is natural to try using the alternating series test. However, the terms do not approach 0 as n goes to infinity, so the alternating series test does not apply. We can instead use the integral test:

Integrating the function f(x) = 1/x ln(x), we have

∫(1 to ∞) 1/x ln(x) dx = lim(t->∞) [ln(ln(t)) - ln(ln(1))]

The limit does not exist, so the integral diverges, and therefore the series is also divergent by the integral test.

Determine If Is Absolutely Convergent, Conditionally Convergent or Divergent