Why does aperys constant converge

The following table summarizes progress in computing upper bounds on the irrationality measure for. Here, the exact values for is given by. Beukers's integral is given by. This integral is closely related to using the curious identity. Sums related to include. Huvent in Gourevitch and rediscovered by B. Cloitre pers. The above equations are special cases of a general result due to Ramanujan Berndt Determining a sum of this type is given as an exercise by Bailey et al. A beautiful double series for is given by.

Oloa, pers. The characteristic polynomial has roots , so. This can be rewritten as. Integrals for include. Amdeberhan used Wilf-Zeilberger pairs with. For ,. Boros and Moll , p. Amdeberhan The corresponding for and 2 are. Gosper expressed as the matrix product. Given three integers chosen at random, the probability that no common factor will divide them all is.

Haible and T. Papanikolaou computed to digits using a Wilf-Zeilberger pair identity with. Amdeberhan and Zeilberger The record as of Dec. Amdeberhan, T. Combinatorics 3 , No. Combinatorics 4 , No. Bailey, D. Experimental Mathematics in Action.

Preprint dated Feb. Berndt, B. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, Beukers, F. London Math. Number Th. Boros, G. Cambridge, England: Cambridge University Press, Borwein, J. New York: Wiley, Castellanos, D.

Part I. Conway, J. New York: Springer-Verlag, pp. Derbyshire, J. New York: Penguin, pp. Dvornicich, R. Amsterdam, Netherlands: North-Holland, pp. Ewell, J. Monthly 97 , , Finch, S. Cambridge, England: Cambridge University Press, pp. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. If there is any sum formula, could you give me the intuition behind this? I really can't just accept formulas, would be great if I could have a explanation. And about the proof of it being irrational, does someone knows an intuitive link that can explain me this?

I only find links that accept things that I don't know. And as I'm asking this only for studying and not for homework, I really wanted to understand as great as possible. It is known that the given series converges very slowly. There are even faster convergent series, obtained by techniques of convergence acceleration. Those who listened casually, or who were afflicted with being non-Francophone, appeared to hear only a sequence of unlikely assertion.

My tentative explanation is based on the following fact. An informal report , Math. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Asked 8 years, 6 months ago. Active 8 years, 5 months ago.