5: You should get

This is still not a series that we would know, the Root and the Ratio tests would still yield 1, it would be nice to get rid of that logarithm. We know that we can estimate it from above by powers, but one should be careful here. The estimate we have here is an upper estimate, which is only good if the resulting test series is convergent. However, if we used, say, ln(k) < k, then the resulting series on the right would have terms 1/k and therefore it would diverge. Thus one has to use smaller upper estimate for logarithm, so that there are "more powers" left in the denominator. For instance, the following should work.

What do you know about convergence of the test series on the right?

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