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If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...

Question I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...

Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...

Questions $$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$ Also obeys (see background for argument): $$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...

There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...

I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...