Exploring the Finiteness of a Sequence with u Variables

[zetafunction]

is the following sequence finite

$$\sum_{n=1}^{\infty} \frac{log^{u-1} (n)}{n} - u^{-1}log^{u}(n)$$

if u=1 then we have simply the Euler-Mascheroni constant but what happens in other cases or other values for 'u'

 

[CRGreathouse]

I'm going to go out on a limb and guess that when you write
$$\sum_{n=1}^{\infty} \frac{log^{u-1} (n)}{n} - u^{-1}log^{u}(n)$$
you mean something like
$$\lim_{x\to\infty}-(\log x)^u+\sum_{n=1}^x\frac{(\log n)^{u-1}}{n}$$
but you may mean something else entirely.

 

[zetafunction]

no but thanks by the answer i meant

$$\sum_{n=1}^{\infty} \frac{log^{u-1}(n)}{n} - \int_{1}^{\infty}\frac{log^{u-1}(x)}{x}$$

in case u=1 we have the Euler Mascheroni constant but how about for other values ??