A New Proof of a Theorem in Analysis by Generating Integrals and Fractional Calculus

The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of Generating Integrals and Fractional Calculus. A generalization of the Riemann zeta function up to non-integer order is derived.