Determine whether the following series converge. Justify your answer.
$$\sum_{j=2}^{\infty} \frac{2}{j \ln^2 j}$$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The Ratio Test yields r = . This is less than 1, so the series converges by the Ratio Test.
(Type an exact answer.)
B. The series is a geometric series with common ratio . This is less than 1, so the series converges by the properties of a geometric series.
(Type an exact answer.)
C. The series is a geometric series with common ratio . This is greater than 1, so the series diverges by the properties of a geometric series.
(Type an exact answer.)
D. Since f(x) = $\frac{2}{x \ln^2 x}$ is continuous, positive and decreasing for x ≥2, and $\int_{2}^{\infty} \frac{2}{x \ln^2 x} dx$ = , the series converges by the Integral Test.
(Type an exact answer.)
E. The Ratio Test yields r = . This is greater than 1, so the series diverges by the Ratio Test.
(Type an exact answer.)
F. Since f(x) = $\frac{2}{x \ln^2 x}$ is continuous, positive and decreasing for x≥2, and $\int_{2}^{\infty} \frac{2}{x \ln^2 x} dx$ = ∞, the series diverges by the Integral Test.
