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## Homework Statement

Determine whether the following series converges absolutely, converges conditionally, or diverges.

[tex]\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3 - ln(n)}[/tex]

## Homework Equations

The assortment of different tests.

## The Attempt at a Solution

Okay, first of all, I tried using Alternating Series Test. This worked and the series satisfied all 3 conditions (decreasing, alternating, and limit as n approaches infinity = 0). This means the

**series must converge either conditionally or absolutely**since I haven't eliminated the possibility of it converging absolutely yet.

So I take the absolute value of the series and if it converges, it's absolutely convergent.

If it diverges, then it's not absolutely convergent. Therefore, that means it's conditionally convergent because I already proved that it must converge in some manner.

So this is

[tex]\sum_{n=1}^{\infty} |\frac{(-1)^n}{n^3 - ln(n)}|[/tex]

[tex]=\sum_{n=1}^{\infty} |\frac{1}{n^3 - ln(n)}|[/tex]

Now how do I find that this series converges or diverges? I tried every test I'm aware of and each was inconclusive. I tried WolframAlpha and it said that the tests were inconclusive, but it gave a number.

**Does this mean it absolutely converges? If so, how would I show my work?**

I tried using the Direct Comparison test and compared the series with the absolute values to 1000/n^3. However, I'm not entirely sure if 1000/n^3 is greater than the absolute value series for all n terms.

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