[Work shown] Which test should I use for this series

Sum from 1 to infinity of (kln(k))/(k+1)^3

I used a Limit Comparison Test, but I believe that it's inconclusive.

a_n is the original series.
b_n is 1/(k+1)^3, which is a convergent series (I used a Comparison Test to show this, comparing it against 1/k^3, a convergent p-series)

So using the Limit Comparison Test:

lim [kln(k)/(k+1)^3] / [1/(k+1)^3], I got:

lim kln(k) = Infinity.

So the series being compared against is convergent, but the test goes to infinity, and that's why I think it's inconclusive.

However, I don't know what other test to use. Integration seems like it would be a lot of work (I'm thinking integration by parts, but there's too many "parts").

Ratio/Root test doesn't look like it would work, and I have tried ratio test already. and it gets complicated.

Can somebody please help?

Thank you.

You have the right idea; just keep the terms as simple as possible.

Using the limit comparison test with Σ k ln k/k^3 = Σ (ln k)/k^2 yields
lim(k→∞) [k ln k/(k+1)^3] / [ln k / k^2] = 1.

So, Σ k ln k/(k+1)^3 converges <==> Σ (ln k)/k^2 converges.

However, Σ (ln k)/k^2 does converge, by the Direct Comparison Test, because ln k < k^(1/2)
for sufficiently large k, and Σ k^(1/2)/k^2 = Σ 1/k^(3/2) is a convergent p-series.

Hence, the series in question converges.

I hope this helps!