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I'm working on the series

sigma(k=1, inf) 1/(lnk)^3

The first thing I wanted to try was to compare it with a similar series, but I couldn't find anything that suited it well.

Next I thought about the limit comparison test, but again I didn't have anything suitable.

Finally, and perhaps the best option, I wanted to try the integral test for f(x) = 1/(lnx)^3 but I had no idea how to integrate that.

Any tips on how one should begin a problem of this nature?

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One way to do it would be to compare with the series sigma (1/k).

The point is that (lnx)^{3} is less than x from some point on. That implies that 1/(lnx)^{3}>1/x from some point on, and so 1/(lnk)^{3}>1/k from some point on. Since sigma (1/k) diverges to infinity, so does sigma 1/(lnk)^{3}.

Now, showing (lnx)^{3}<x for large enough x is the same as showing lnx<x^{1/3} for large enough x, and that could be done using calculus 1 by analyzing that function using derivatives.