It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I can't remember where I heard this) that the harmonic...

Consider the regulator $$R(n,s) = \frac{n^s + n^{-s}}{2}$$ Note that $\lim_{s \rightarrow 0} R(n,s) = 1$. Then \begin{align} \sum_{n=1}^\infty n^{-1} &= \sum_{n=1}^\infty n^{-1} \lim_{s \rightarrow 0} R(n,s) \\ &= \sum_{n=1}^\infty \lim_{s \rightarrow 0} n^{-1} R(n,s) \\ &\overset{\star}{=} \l...