We develop new closed form representations of sums of quadratic alternating harmonic numbers and reciprocal binomial coefficients.

We extend some results of Euler related sums. Integral and closed-form representations of sums with products of harmonic numbers and binomial coefficients are developed in terms of Polygamma functions. The various representations presented in this paper are believed to be new.

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An integral inequality is developed from which when applied to composite quadrature rules in numerical integration it is proved that there is a three fold improvement in the remainder of the classical averages of the Midpoint and Trapezoidal quadratures. Inequalities for special means are also given.