Open problems in infinite series and integrals
问题内容
I am curious about infinite series and definite integrals for which the closed-form evaluation is currently an open problem.
Now, one can of course think of some strange, complicated infinite series like
$$ S:= \sum_{n=1}^{\infty} \frac{1}{n^{n!}} \ \ , $$ but I am not interested in such series.
Instead, I am looking for series that appear in the mathematical literature in some form. For instance, the following three rational zeta are listed here in equations $(132)$, $(136)$, and $(140)$, respectively as having no closed-form expressions:
\begin{align*} C_{1} &:= \sum_{n=2}^{\infty} \frac{\zeta(n)}{n!} \\ & \approx 1.078188729575818482758265436769832381707219, \\\qquad C_{2} &:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{n!} \\ & \approx 2.407446554790328514709486656223022725582266, \text{ and} \\ C_{3} &:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!} \\ &\approx 0.869001991962908998811054805561395688892494.\end{align*}
Another example is the series
$$ D: = \sum_{n=2}^{\infty} \frac{\zeta(n)-1}{n-1} \ \ , $$ which is described in OEIS A085361 and references therein.
It also relevant to note that I am preferably not looking for well-known examples, like Apéry's constant or the Flint-Hills series.
回答 (1)
Suppose you have a long metal bar. If you twist one end relative to the other, the relation is $$T = GJ \frac{d\theta}{dz}$$ $T$ is applied torque, $G$ is the material's shear modulus, $\frac{d\theta}{dz}$ is twist per unit length, and $J$ is the torsion constant of the cross-section. If the cross-section of the bar is a square of side length $a$, for example, then $$J = \frac{a^4}{3} \left( 1 - \frac{192}{\pi^5} \sum_{\substack{n\ge1 \\ n \text{ odd}}} \frac{\tanh\left(n\pi/2\right)}{n^5} \right)$$ Where the infinite sum has no known closed form. The same goes for various other polygonal cross-sections. You can check out MathWorld for some further examples.