On proofs and generalizations for some formulas in actuarial mathematics

In this paper, we try to give some proofs and generalizations for some formulas in actuarial mathematics. At first, we generalize recursive formulas for reserves at fractional durations for frequently collected premiums in a year. Secondly, under the assumption of uniform distribution of deaths in each year of age, strict proofs are provided for relation formulas to actuarial present values of life annuities with single payment or mthly payments in a year.