Pi: A Source Book pp 112-128 | Cite as

# On the Use of the Discovered Factors to Sum Infinite Series

## Abstract

If 1 + *Az + Bz* ^{
2}
+ *Cz* ^{
3}
+ *Dz* ^{
4}+ ... = (1 + α*z*)(1 + β*z*)(1 + γ*z*)(1 + δ*z*) ..., 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.

## Keywords

Previous Series Ordinary Algebra Equal Exponent## Preview

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