TITLE: Is it possible to use regularization methods on the Harmonic Series?
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: 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).