On 2015-10-31 17:39, Warren D Smith wrote:
Specifically Gosper 1974 found
H(x) = -SUM(j=1,2,3...) (1/j) * (j+x)^(-1) * (3*j+x) * binomial(j-x-1, j) / binomial(2j, j)
where
H(n) = 1 + 1/2 + 1/3 + ... + 1/n.
Each summand in Gosper's series is a rational function of x with denominator=j+x and having numerator which is a polynomial(x) of degree=j+1. With a little cleverness, the first N terms in Gosper's series can all be computed in O(N) arithmetic operations.
My question was, what if anything is the corresponding formula for 1 + 1/3 + 1/5 + ... + 1/(2*n-1).
At least two things. E.g., you could subtract the even ones from the bunch. But that's not much of a correspondence. What I suggested was to start n0 at 1/2 instead of 1, 1/(1/2) + 1/(3/2) + 1/(5/2) ... = binomial stuff with n0 = 1/2 instead of 1. But it looks like you'll need to plug in x+1/2 instead of x as well. Gaa, that pre-matrix viewpoint was messy. Tomorrow night I'll try to modernize (a la http://gosper.org/pathi.pdf [1]) it so you'll have a general formula, including 1,4,7,10, or whatever. --rwg Links: ------ [1] http://gosper.org/pathi.pdf