Can the hyperreals be used to describe the "gold nuggets" found with nonconverging series and Casimir forces? In one of Numberphile's videos they describe the $-\frac{1}{12}$ as a "gold nugget" inside the sequence $1+2+3+4+\dots$ surrounded by a bunch of "rock" that is infinity.
Can these numbers that we get with regularization/analytic continuation of Reimann zeta/etc be described more simply using the hyperreals? Is there some particular non-arbitrary unlimited hyperreal $A$ you can associate with $1+2+3+4\dots = A-\frac{1}{12}$? Like in a Casimir force calculation problem, where the forces on both sides are infinite but differ by some finite value (related to the $-\frac{1}{12}$), is there some specific, non-arbitrary unlimited hyperreal value $A$ associated with the infinite part of that force? 
 A: No. Robinson's hyperreals just are a different way of getting at the same results that you could get with traditional $\varepsilon$-$\delta$ definitions. Since the sum traditionally diverges, that's what hyperreals tell you.
Now, you could probably formalize your favorite way of getting to $-1/12$ with the hyperreals in place of whatever limits you'd calculate, but that would not be a method at getting the value "more simply".
A: The reason the expression $\sum_{i=0}^\infty a_i$ makes sense, is because if it converges, the point of convergence is unique. 
So, for any converging series $(a_i)_{i\in\mathbb{^*N}}$ we also have $\sum_{i=0}^{h_1} a_i \approx \sum_{i=0}^{h_2} a_i $ , for $h_1,h_2\in\mathbb{^*N-N}$.  However, if a series is diverging, then $\sum_{i=0}^{h_1} a_i  - \sum_{i=0}^{h_2} a_i $ can have infinitely many results depending on your choice of $h_1,h_2$. 
So while for every $h\in\mathbb{^*N-N}$ we can find an $A\in \mathbb{^* R}$ so that $\sum_{i=0}^{h} i = A-\frac 1 {12}$ holds, there can be no constant $A$ that fulfills this equation for all possible $h\in\mathbb{^*N-N}$.
While infinite and infinitesimal values in nonstandard analysis exist, their nature is rather analytical. As we can't construct explicit infinite values, their main use is that all finite values from their perspective "look the same". As example: For any infinite number $h$ and any finite number $n$, the equation $\frac n h \approx 0$ holds.
So, to pick up your example of Casimir-force: Whether we are in standard- or nonstandard-analysis, we don't actually want to discriminate infinite values, we merely want to show and use the fact that these infinite values are greater than any finite value. 
So, if we had two infinite sums whose difference is finite, in both standard- and nonstandard-analysis we'd look at the difference of both sums rather than at each sum individually.
