Closed form for (in)finite sums

[henpen]

I have a set of questions concerning the perennial sum
\large \sum_{k=1}^{n}k^p
and its properties.

1. For p \ge 0, the closed form of this is known (via Faulhaber's formula).
I know little about divergent series, but I've read that in some sense there exists a value associated with these sums as n \rightarrow \infty: for instance \lim_{n \rightarrow \infty}\sum_{k=1}^{n}k = \frac{-1}{12} (here)? I think this is called zeta function regularisation.

a. What is the significance of such sums? Are they useful realistic compatable-with-rest-of-mathematics, or wholly artificial? What evidence is there of these pseudosums' existence?

b. What are they for p \ge -1 (hoping that the Harmonic series, p = -1, has such a 'pseudo-sum')?2. Moving on to p < 0. For even p the limit as n \rightarrow \infty infinity is known (Euler).

a. Is there a Faulhaber equivalent in this case (a closed form of the sum up to n for either even or integer n)?

b. Are the odd p's infinite series likely to be found?For comparison, with constant k and changing p:
\large \sum_{p=1}^{n}k^p= \frac{1-k^{p+1}}{1-k}
I would love if for constant p and changing k was a formula as complete as this.

Thank you for volunteering to suffer my question.

 

[fresh_42]

The problem why this can't be is a dimensional one. If we have a constant $p$ and changing $k$ as in $\sum_{k=1}^n k^p$ we basically add different dimensions:
$p=1$ is adding lengths
$p=2$ is adding areas
$p=3$ is adding volumes
etc.
So there cannot be one formula fits all, because the procedure is so different, and with increasing dimension $p$ the additional amount of each term is less and less relevant compared to the partial sum so far.