Is it possible to use regularization methods on the Harmonic Series?

I recently learned about summation methods when dealing with divergent series to give them a finite value. An example of this isusing Cesàro summation on Grandi's series to get 1/2. However every method I know of is unable to sum the harmonic series. Are there any summation methods that work on the harmonic series or is it provably impossible to sum this series?

I'll stick to real, positive exponents $s$ in $\sum_{n=1}^\infty n^{-s}$. The following regularization works for $s\in (0,\infty)\setminus \{1\}$: formally, $$(1-2^{1-s})\sum_{n=1}^\infty n^{-s} = \sum_{n=1}^\infty n^{-s} - 2\sum_{n=1}^\infty (2n)^{-s} = \sum_{n=1}^\infty (-1)^{n-1}n^{-s} \tag1$$ where the right-hand side of (1) converges (by the alternating series test) for $ s>0$. Thus, for $s\in (0,\infty)\setminus \{1\}$ we can interpret $\sum_{n=1}^\infty n^{-s} $ as $$\zeta (s)=(1-2^{1-s})^{-1}\sum_{n=1}^\infty (-1)^{n-1}n^{-s} \tag2$$ Although setting $s=1$ in (2) does not work, we can do simple averaging: $$\sum_{n=1}^\infty n^{-1} \,``=" \lim_{\epsilon\to 0} \frac{\zeta(1-\epsilon)+\zeta(1+\epsilon)}{2} =\gamma \tag3$$ where $\gamma$ is the Euler-Mascheroni constant, $\gamma=0.57721\dots$ Indeed, the averaging in (3) cancels out the contribution of the simple pole in the Laurent series $$\zeta(s)=\frac{1}{s-1}+\gamma+O(s-1)$$

(Adapted from MathOverflow).