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

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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]:

an = nth Left Riemann Sum with uniform partition

bn = nth Right Riemann Sum with uniform partition

cn = nth Midpoint Riemann Sum with uniform partition

dn = nth Trapezoidal Riemann Sum with uniform partition

Note that, by definition, 2dn = an+bn

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 Cω function?
 
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