Finding conditions for monotone convergence of a particular sequence. (1 Viewer)

[Paradoxica]

Suppose f is a non-positive XOR non-negative, singularity free, smooth, C^∞ function on the interval [0,1].

Define the following sequence of numbers, which approximate the integral of f on [0,1]:

a_n = n^th Left Riemann Sum with uniform partition

b_n = n^th Right Riemann Sum with uniform partition

c_n = n^th Midpoint Riemann Sum with uniform partition

d_n = n^th Trapezoidal Riemann Sum with uniform partition

Note that, by definition, 2d_n = a_n+b_n

What are necessary and sufficient conditions such that all four sequences monotonically converge to the integral of f?

Do the conditions change if f is a C^ω function?