Convergence of the summation formulas constructed by using a symbolic operator approach

This paper deals with the convergence of the summation of power series of the form Sb a(f ;x) = ∑ a≤k≤b f(k)x k, where 0 ≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) a differentiable function defined on [a, b). Here the summation is found by using the symbolic operator approach shown in [4] . We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Some examples such as the generalized Euler’s transformation series will also be given. In addition, we will compare the convergence of the given series transforms. AMS Subject Classification: 65B10, 39A70, 41A80, 05A15. ∗The research of this author was partially supported by Applied Research Initiative Grant of UCCSN 1 2