math help, the given series is absolutely convergent, conditionally convergent or divergent:

Question

Determine whether the given series is absolutely convergent, conditionally convergent or

divergent:

∑(-1)^n ln((n)^2)/ln^(2)(n)



from n = 2 to infinity

Answer 1

∑_n=2^∞(-1)ⁿln(n²)/(ln²n).

1. lim n→∞In(n²)/(ln²n) = lim n→∞ 2/lnn = 0;
2. ln(n + 1)²/ln²(n + 1) = 2/ln(n + 1) < 2/lnn = lnn²/ln².

^{So, by alternative series test our test convergent.}

^{But absolute values series ∑}n=2^∞ln(n²)/(ln²n)diverges becouse by comparison test ln(n²)/ln²n = 2/lnn > 2/n

^{and series .}∑_n=2^∞2/n diverges.

So, given series converges conditionally.