I tried Mathematica, various regularization methods, including Borel, with no result.
On Math.SE the question was attacked with claims that divergent series cannot have a sum, so I decided to ask at a site of higher level.
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$\begingroup$ For what range of $s$? For $s=0$, the sum is $\sum _{x=1}^{\infty } i$. $\endgroup$– Joseph O'RourkeJan 31, 2018 at 21:30
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2$\begingroup$ The now closed cross-post has one answer: math.stackexchange.com/questions/2630266/… $\endgroup$– jeqJan 31, 2018 at 23:34
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1$\begingroup$ Anixx, could you clarify the context of your question? Does this sum arise in some physics context? Why would one expect a meaningful answer? $\endgroup$– Mikhail KatzFeb 1, 2018 at 10:16
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$\begingroup$ Noting the symbolic series $f(s):=\sum_{x\geq1}\sqrt{s^2x^2-1}$ and the meromorphic function $g(s):=\sum_{x=1}^{\infty}\frac{1}{\sqrt{s^2x^2-1}}$, I'd use the (formal) functional relation $sf'(s)-f(s)=g(s)$ to obtain a regularization: $f(s)=s(c+\int_{u=\infty}^s \frac{g(u)}{u}du$) for some constant. Inverting integration and summation in $g$ yields a family of integrals which I fear is not solvable by elementary functions (I'd say elliptic integrals or similar). Don't know if there are computable recombination. Note that the regularization $f$ may well be multivalued... $\endgroup$– Loïc TeyssierFeb 1, 2018 at 15:43
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1$\begingroup$ @Mikhail Katz Actualy, my final aim is to find the generalized vialue of the divergent integral $\int_1^\infty \sqrt{x^2ā1}dx$ (area of the hyperbola). For that I need to find limit $\lim_{sā0}sā$ where ā is the generalized value of this sum. $\endgroup$– AnixxFeb 2, 2018 at 4:26
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