Timeline for What is the generalized sum of the following series? $\sum _{x=1}^{\infty } \sqrt{s^2 x^2-1}$ [closed]
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2018 at 3:01 | review | Reopen votes | |||
| Feb 6, 2018 at 10:12 | |||||
| Feb 2, 2018 at 8:08 | comment | added | Mikhail Katz | Why isn't the answer at MSE satisfactory? | |
| Feb 2, 2018 at 4:26 | comment | added | Anixx | @Mikhail Katz Actualy, my final aim is to find the generalized vialue of the divergent integral $\int_1^\infty \sqrt{x^2−1}dx$ (area of the hyperbola). For that I need to find limit $\lim_{s→0}s∑$ where ∑ is the generalized value of this sum. | |
| Feb 1, 2018 at 15:49 | comment | added | Loïc Teyssier | ... Hence the choice of the constant $c$ is not unique (and not real in general). A canonical choice would be $c:=\zeta(-1)=-\frac{1}{12}$ due to the asymptotics $s\to+\infty$. | |
| Feb 1, 2018 at 15:43 | comment | added | Loïc Teyssier | 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... | |
| Feb 1, 2018 at 10:16 | comment | added | Mikhail Katz | Anixx, could you clarify the context of your question? Does this sum arise in some physics context? Why would one expect a meaningful answer? | |
| Feb 1, 2018 at 5:50 | review | Reopen votes | |||
| Feb 1, 2018 at 12:03 | |||||
| Feb 1, 2018 at 3:17 | history | closed |
Andrés E. Caicedo David Handelman Chris Godsil Gerald Edgar Pace Nielsen |
Needs details or clarity | |
| Jan 31, 2018 at 23:34 | comment | added | jeq | The now closed cross-post has one answer: math.stackexchange.com/questions/2630266/… | |
| Jan 31, 2018 at 21:53 | review | Close votes | |||
| Feb 1, 2018 at 3:17 | |||||
| Jan 31, 2018 at 21:30 | comment | added | Joseph O'Rourke | For what range of $s$? For $s=0$, the sum is $\sum _{x=1}^{\infty } i$. | |
| Jan 31, 2018 at 21:14 | history | asked | Anixx | CC BY-SA 3.0 |