A pretty series revisited

In the May 1954 issue of the Gazette Daniel F. Ferguson challenged readers to devise their own proof for what he described as a curious and somewhat pleasing sum (see [1])

Monotonicity of the midpoint and trapezium estimates for integrals

The ‘midpoint’ approximation to the integral $$\int_0^1 f $$ is $${M_n}\left( f \right) = {1 \over n}\sum\limits_{r = 1}^n f \left( {{{2r - 1} \over {2n}}} \right)$$ .