P.S. The puzzle I just posted suggested another one (new to me), of which I have no idea of the answer, or more accurately have too many things to do right now, so can't think about it. (Posting to math-fun is a very successful technique for procrastinating.) Suppose the integer coefficients c_j of this infinite series: (***) x = c_1/1 + c_2/2 + c+3/3 + . . . + c+k/k + . . . are periodic (for some n in Z+ and all j in Z+,, c_j = c_(j+n)), and that the series converges. The c_j are otherwise arbitrary. Can x be evaluated? --Dan P.S. As may be obvious, I had too much coffee this morning.
On Feb 11, 2015, at 12:57 PM, Daniel Asimov <asimov@msri.org> wrote:
On Feb 11, 2015, at 11:20 AM, Daniel Asimov <asimov@msri.org> wrote:
(**) f = c_1 / 2! + c_2 / 3! + . . . + c_k / k! + . . . with 0 <= c_j <= j for all j
This was in the American Mathematical Monthly ca. 1964, and just happened to be something I was working on at the time:
PUZZLE:
Suppose that in (**) the coefficients c_j are periodic: For some n and for all j,
c_(j+n) = c_n
.
Evaluate the fraction f in terms of c_1, . . ., c_n.