Everyone knows that the n-th triangular number squared is the sum of the first n cubes, but do you know that the n-th triangular number cubed divided by the sum of the first n fifth powers tends to 0.75?
This (and all the rest that you mention) are just facts about ratios of leading coefficients of same-degree polynomials. The sum of the first n k'th powers is some polynomial, whose lead coefficient is n^(k+1)/(k+1). There are lower-order terms because you're doing a discrete approximation to integration. If you want to know more about this, read up on Bernoulli polynomials. The triangular number t_n is the sum of first powers, so the leading term is n^2/2. The leading term of (t_n)^r is n^(2r)/(2^r). So (t_n)^r / (sum of the first n 2r-1'th powers) will tend to 2r/(2^r). This gives 1, 1, 3/4, 1/2, 5/16, 3/16, 7/64, 1/16, 9/256, 5/256 for r=1,2,3,.. --Michael Kleber kleber@brandeis.edu