Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?
Letting $$ H=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots $$ and assuming convergence, we have $$ 0=H-2\cdot\frac{1}{2}H=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots\right) - 2\cdot\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\ldots\right)=\\1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\ldots=\ln 2. $$ Since this is a contradiction, the sum must diverge.
Update.
To be more clear about what I'm saying here... I'm not saying that there's no way to assign a value to the harmonic series. Clearly there are any number of ways, some sillier and more arbitrary than others. Call a partial function $S$ from infinite sequences to real or complex numbers a summation method if $S(\{a_i\})$ is defined and equal to $\sum_{i=1}^{\infty}a_i$ whenever the sum converges (in the usual sense); and say that $\{a_i\}$ is $S$-summable if $S(\{a_i\})$ is defined. There are a number of properties that you might want your summation method to preserve from ordinary summation. For instance, you might want it to remain linear: $$S(\{ \alpha a_i + \beta b_i \})=\alpha S(\{a_i\}) + \beta S(\{b_i\})$$ whenever $\{a_i\}$ and $\{b_i\}$ are $S$-summable. You might also want it to remain stable under insertion of zeros: if $J:\mathbb{N}\rightarrow\mathbb{N}$ is increasing and $\{a_i\}$ is $S$-summable, then $S(\{a'_i\})=S(\{a_i\})$, where $a'_{J(i)}=a_i$ and $a'_i=0$ for $i\not\in J^{-1}(\mathbb{N})$. I haven't said anything about analytic continuation or whatever other exotic regularization you want to consider. But what I'm saying, above and in the comments, is that no summation method under which either the harmonic series or $1+2+3+\ldots$ is summable can be both linear and stable.
Now, there's a weaker form of stability (under insertion of only finitely many zeros) that you can preserve while making the harmonic series summable, but not while making $1+2+3+\ldots$ summable. So in that specific sense, the latter series (!) is more pathological. In order to sum the natural numbers, you need to give up either linearity or stability, rendering the manipulations in the original post meaningless.
You should note that the Cauchy principlal value of $\zeta(1)$ is $\gamma$ (Euler-Mascheroni constant):
$$\lim_{h\to0}\frac{\zeta(1+h)+\zeta(1-h)}2=\gamma$$
The same value can be obtained using the Ramanujan's summation of harmonic series.
So to directly answer your question, yes, we can assign a value to the sum of harmonic series, and at least the two methods give the same result.
Note that the Ramanujan's sum of $1+2+3+4+...$ is $-\frac1{12}$ so this method can be seen to work in both cases.
For anyone who happens upon this question and doesn't understand, like I did, I would like to elaborate on exactly what I specifically was missing in my understanding.
When I posted this question, I didn't know what people meant when they said $\sum_{n=1}^\infty n = -1/12$. Now that I'm older and a bit more experienced, this statement is mathematically wrong. Equating these two quantities is simply a false, pseudo-mathematical inanity. No one told me this when I heard about it, so I always perceived it as mathematical fact.
So what do people actually mean when they write that? Well, we can define the Riemann Zeta Function: $$ \zeta(s) = \sum_{n=1}^\infty\frac1{n^s} $$ for a complex number $s$. We know that this sum converges for $\Re(s)>1$, and it diverges when $s=1$, and $\Re(s)<1$. However, quite frequently we talk about values of the zeta function for which $\Re(s)<1$. This is because we're not really talking about the zeta function, but rather an (and the unique) analytic continuation of the zeta function.
To be specific, if a function $f$ is analytic in a domain $U$, and $V\supset U$, and $F$ is some function analytic on $V$ such that $F(z)=f(z)$ for $z\in U$, then $F$ is an analytic continuation of $f$ to a larger domain. It turns out that this function $F$ is unique. Since the zeta function is analytic in the half-plane $\Re(s)\ge 1$, minus the point $s = 1$, it has an analytic continuation to $\mathbb{C}\backslash\{1\}$, which is uniquely determined by the behaviour of the function $\zeta(s)$ in the domain of convergence of the infinite sum.
One may be curious as to why this analytic continuation cannot be "infinite everywhere" (which would seem to be the only logical choice for such a function given that the sum doesn't converge anywhere outside $\Re(s)>1$). If this were true, then the reciprocal of that analytic continuation would be identically zero on a dense subset of the complex plane, which, by the principle of analytic continuation once again, must imply that the function is identically zero everywhere. But this doesn't make sense since $1/\zeta(s)$ is clearly defined for $\Re(s)>1$.
Hence, there is some meaning to the values $\zeta(s)$ for $\Re(s)<1$, but this does not mean that $\zeta(s) = \sum_{n=1}^\infty\frac1{n^s}$ for all $s$. The sum diverges. The zeta function is extended.
However..., if we abuse our notation a bit, and let $\sum_{n=1}^\infty\frac1{n^s}$ denote the analytic continuation, rather than the actual sum, for $s$ outside the domain of convergence, then we can plug in $s = 1$ to get $$ \zeta(-1) "=" 1 + 2 + 3 + 4 + \cdots $$ and one can prove, properly (i.e. using real math, not pseudo-math), that $\zeta(-1)$ (that is, the analytic continuation of $\zeta$ evaluated at $s=-1$), is indeed equal to $-1/12$.
As for the sum $\sum_{n=1}^\infty\frac1{n}$, this is simply a pole of the Riemann zeta function. The zeta function, on the entire real line, is a meromorphic function (i.e. it is holomorphic/analytic everywhere except on a countable set with no limit points). Functions like these are "allowed" to have singularities on a countable number of disjoint points. They can be thought of as parts where the function "looks like" $1/(s-z)^k$ for some $k$. (There are also "essential singularities" which go to infinity faster than this for any $k$, but the pole of the zeta function is not essential).
Essentially, what this means is that $\sum_{n=1}^\infty n = -1/12$ doesn't contradict some notion of convergence/divergence which implies $\sum_{n=1}^\infty\frac1n = \infty$. Both diverge, we just have a "special way" of evaluating the former sum.