Truncating $p$-series Let $p \in \mathbb{R}$. Series of the form $$S_p = \sum_{k \in \mathbb{N^*}} \frac{1}{k^p}$$ converge if and only if $p > 1$. Let us define $$S_{p, n} \triangleq \sum_{k = 1}^{n} \frac{1}{k^p},$$ the truncation of the $p$-series at its $n$-th term. (It is evident that $S_{p, n} \space \xrightarrow{n\to\infty} \space S_p$.) Is there a closed expression for $S_{p, n}$, at least for some values of $p$ (integers must be easier)?
 A: In Maple notation, for $p \ge 2$ integer, $$S_{p,n} = \zeta(p) + \frac{(-1)^{p-1}}{(p-1)!} \Psi(p-1,n+1),$$ where $\psi(m,\space·)$ is the $m$-th polygamma function.
A: Yes, there is.  These are called the generalized harmonic numbers.  
With polygamma function: I suggest looking at this paper, they have many interesting things.  In particular, if we define 
$$
H_n(z;r)=\sum_{k=1}^n\frac{1}{(z+k-1)^r}
$$
(even more generalized) then we have for integers $p$ 
$$
H_n(z;p)=\frac{(-1)^{p-1}}{(p-1)!}(\psi^{(p-1)}(z+n)-\psi^{(p-1)}(z))
$$ 
where 
$$
\psi^{(k-1)}(z)=\frac{d^k}{dx^k}\left(\log\Gamma(x)\right) \biggr|_{x=z}.
$$
This is equation (1.14) of the paper, but I think they forgot a factor of $(p-1)!$ on the denominator. Taking the case $z=1$ gives what you are looking for.
With zeta function: we can write $$H_n(r)=\sum_{k=1}^n\frac{1}{k^r}=\zeta(r)-\zeta(r,n+1),$$ where $\zeta(s,a)$ is the Hurwitz zeta function. This $\space$ really clear, but by using identities regarding $\zeta(s,a)$ we can change the identity into $\space$ previous one, and other things.
Hope that helps.
A: To the best of my knowledge, there is no closed form expression for $S_{p,n}$. 
EDIT: Let me amend my answer to read, to the best of my knowledge, there is no closed form expression in terms of functions familiar to, say, students of 1st-year Calculus. 
