On the Use of the Discovered Factors to Sum Infinite Series

Abstract

If 1 + Az + Bz ² + Cz ³ + Dz ⁴+ ... = (1 + αz)(1 + βz)(1 + γz)(1 + δz) ..., then these factors, whether they be finite or infinite in number, must produce the expression 1 + Az + Bz ²+ Cz ³+ Dz ⁴+ ..., when they are actually multiplied. It follows then that the coefficient A is equal to the sum α + β + γ + δ + ε + .... The coefficient B is equal to the sum of the products taken two at a time. Hence B = αβ + αγ + αδ + βγ + βδ + γδ + .... Also the coefficient C is equal to the sum of products taken three at a time, namely C = αβγ + αβδ + βγδ + αγδ + .... We also have D as the sum of products taken four at a time, and E is the sum of products taken five at a time, etc. All of this is clear from ordinary algebra.