A math thing I worked out
It's somehow been in the air in a few different math places I visit online what the sum of:
and
are. Among other things, these series come up when working out the probability that with randomly chosen values for x and y in the range (0, 1) whether floor(x/y) is even or odd, or whether round(x/y) is even or odd.
Now, these are very well known series and the quite well-known answer to these are:
The usual way that I've seen to solve the first of these () involves showing somehow that it's the Taylor series for evaluated at ; this can be justified in detail or presented “out of thin air”. The usual way to solve the second of those involves rewriting it as:
and then working that out. For some reason these two series are approached completely separately as though the solution for one has absolutely nothing to do with the other, even though the series are obviously closely related. (I'll grant that the answers, at least for now, look completely unrelated)
Anyway, I stumbled on a way of looking at these series that makes it obvious how the two series above are related, and that as far as I can tell easily extends to the much more general case of:
( More math than you probably wanted to read today )