A theoretical summation-integral scheme involving commutation function in life insurance business

The aim of this paper is to analytically extend Euler’s summation-integral quadrature to core actuarial functions based on sound judgement of numerical analytics. Specifically, the objectives are to theoretically (i) Obtain the value of a continuous commutation function relating the value of a discrete sum to its integral (ii) Estimate the n-term life annuity due based on the estimated force of mortality and force of interest using alternative mathematical technique (iii) Estimate the temporary life expectancy based on the estimated force of mortality and force of interest. One of the most relevant applications of this paper is to provide a sound estimate of commutation functional values used in life and pension funds valuation. We used the elementary summation-integral formula of Euler-Maclaurin to relate the value of a discrete sum  _t=n^kƩc+x to its integral ^ω−x_t=n∫Cx+t dt in terms of the derivatives of a continuous commutation function (C=x,t) at two distinct points a and b specified in the integral and a remainder term (R=x,t). We assume the remainder term (R=x,t)→ o(1) where ₀(1)is small tending to zero as b grows very large and this can be used to compute asymptotic expansions for sums.