On the Use of the Discovered Factors to Sum Infinite Series

Abstract

If EquationSource?% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU % caRiaadgeadaWgaaWcbaGaamOEaaqabaGccqGHRaWkcaWGcbWaaSba % aSqaaiaadQhaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaam % 4qamaaBaaaleaacaWG6baabeaakmaaCaaaleqabaGaaG4maaaakiab % gUcaRiaadseadaWgaaWcbaGaamOwaaqabaGcdaahaaWcbeqaaiaais % daaaGccqGHRaWkcqWIVlctcqGH9aqpdaqadaqaaiaaigdacqGHRaWk % cqaHXoqycaWG6baacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaey4kaS % IaeqOSdiMaamOEaaGaayjkaiaawMcaamaabmaabaGaaGymaiabgUca % Riabeo7aNjaadQhaaiaawIcacaGLPaaadaqadaqaaiaaigdacqGHRa % WkcqaH0oazcaWG6baacaGLOaGaayzkaaGaeS47IWeaaa!61C9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1 + Az + B{z^2} + C{z^3} + D{z^4} + \cdots \ = \left( {1 + \alpha z} \right)\left( {1 + \beta z} \right)\left( {1 + \gamma z} \right)\left( {1 + \delta z} \right) \cdots $$, then these factors, whether they be finite or infinite in number, must produce the expression EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU % caRiaadgeacaWG6bGaey4kaSIaamOqaiaadQhadaahaaWcbeqaaiaa % ikdaaaGccqGHRaWkcaWGdbGaamOEamaaCaaaleqabaGaaG4maaaaki % abgUcaRiaadseacaWG6bWaaWbaaSqabeaacaaI0aaaaOGaey4kaSIa % eS47IWeaaa!46FD!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1 + Az + B{z^2} + C{z^3} + D{z^4} + \cdots $$, when they are actually multiplied. It follows then that the coefficient A is equal to the sum EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey % 4kaSIaeqOSdiMaey4kaSIaeq4SdCMaey4kaSIaeqiTdqMaey4kaSIa % eyicI4Saey4kaSIaeS47IWeaaa!445C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\alpha + \beta + \gamma + \delta + \in + \cdots $$. The coefficient B is equal to the sum of the products taken two at a time. Hence EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2 % da9iabeg7aHjabek7aIjabgUcaRiabeg7aHjabeo7aNjabgUcaRiab % eg7aHjabes7aKjabgUcaRiabek7aIjabeo7aNjabgUcaRiabek7aIj % abes7aKjabgUcaRiabeo7aNjabes7aKjabgUcaRiabl+Uimbaa!529F!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$B = \alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta + \cdots $$. Also the coefficient C is equal to the sum of products taken three at a time, namely EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2 % da9iabeg7aHjabek7aIjabeo7aNjabgUcaRiabeg7aHjabek7aIjab % es7aKjabgUcaRiabek7aIjabeo7aNjabes7aKjabgUcaRiabeg7aHj % abeo7aNjabes7aKjabgUcaRiabl+Uimbaa!50DC!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$C = \alpha \beta \gamma + \alpha \beta \delta + \beta \gamma \delta + \alpha \gamma \delta + \cdots $$. 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.