On Apr 3, 5:50 pm, Paul <pepste...@gmail.com> wrote: > All real numbers are obviously either rational, (algebraic and irrational) or transcendental. So these descriptors partition the reals into three disjoint sets. > > From what I've seen, all the difficult results placing a real number into one of these classes have been of the form "x is transcendental". > > Does anyone know of any non-trivial results which show that a specific number is rational or algebraic? In other words, does anyone know any non-trivial results (advanced undergraduate or higher) which define a specific x and then prove a statement of the type "x is rational" or "x is algebraic"?
Let S(n) = sum (1 / (pi*k)^n) over 1 <= k < infinity.
For some n, S (n) is proven rational, and for others it isn't. (It is related to Bernoulli numbers when n is even).
For example, the infinite sum of 1/k^2 is pi^2 / 6, so S (2) = 1/6.