I have been working on characterizing the asymptotic behavior for large $n$ for the following sum:


$$\Gamma(1+t)\sum\_{k=1}^n \frac{\binom{n}{k} (-1)^{k-1}}{k^t}$$


where $t(>0)$ is a given constant. The case for $t=1$ can be somewhat found here:


<http://www.math.wvu.edu/~gould/Vol.4.PDF>


in equation $8.25$. Using this equation, for large $n$, the sum behaves as $\log(n)$. I am trying to understand if similar behavior can be obtained for a general $t>0$.


Thanks in advance.



An asymptotic characterization for a series
Mathematica does the job for concrete values of $t\in \mathbb{N}$:



```
Series[Gamma[1 + 2]*Sum[Binomial[n, k]*(-1)^(k - 1)/k^2, {k, 1, n}], {n, Infinity, 2}]

```


> 
> $$\left(\log ^2(n)+2 \gamma \log (n)+\gamma ^2+\frac{\pi ^2}{6}\right)+\frac{\log (n)+\gamma -1}{n}+\frac{-2 \log (n)-2 \gamma +9}{12 n^2}+O\left(\left(\frac{1}{n}\right)^3\right) $$
> 
> 
> 



```
Series[Gamma[1 + 3]*Sum[Binomial[n, k]*(-1)^(k - 1)/k^3, {k, 1, n}], {n, Infinity, 2}]

```


> 
> $$\left(\log ^3(n)+3 \gamma \log ^2(n)+3 \gamma ^2 \log (n)+\frac{1}{2} \pi ^2 \log (n)+2 \zeta (3)+\gamma ^3+\frac{\gamma \pi ^2}{2}\right)+\frac{6 \log ^2(n)+12 \gamma \log (n)-12 \log (n)+6 \gamma ^2+\pi ^2-12 \gamma }{4 n}+\frac{-6 \log ^2(n)-12 \gamma \log (n)+54 \log (n)-6 \gamma ^2-\pi ^2+54 \gamma -60}{24 n^2}+O\left(\left(\frac{1}{n}\right)^3\right) $$ 
> 
> 
> 


The output for $t=4$ is too long to be cited here. Mathematica fails with $t= \frac 5 2$.


