Let us define $H(k, n) = \displaystyle\sum_{i = 1}^{n} \frac{(\log i)^{k}}{i}$. We want to show that $H(k, n) - \displaystyle\frac{(\log n)^{k + 1}}{k + 1}$ converges as $n \rightarrow \infty$.
Notice that $\displaystyle\int_{1}^{n} \frac{(\log x)^{k}}{x} dx = \frac{(\log n)^{k + 1}}{k + 1}$. Therefore we wish to estimate the difference of the sum and integral. We can use Euler's summation formula to get:
$\displaystyle\sum_{i = 1}^{n} \frac{(\log i)^{k}}{i} - \displaystyle\int_{1}^{n} \frac{(\log x)^{k}}{x} dx = -(t - [t] - c)f(t)|^{n}_{1} + \int_{1}^{n} (t - [t] - c)t^{-2}(k - \log t)\log(t)^{k - 1} dt$
Choosing $c = 0$, the first term on the right side cancels. Since $t - [t] \leq 1$, and $\displaystyle\int_{1}^{n} t^{-2}(k - \log t)\log(t)^{k - 1} dt = \frac{\log(n)^{k}}{n}$, the integral on the right vanishes as $n \rightarrow \infty$ and the difference is... zero?
Obviously this is false but I can't see what's wrong with the argument. The formula for Euler's summation I found in Bateman's Analytic Number theory, page 47. Can anyone find the error here?