We have used the annotated bibliography of proofs of the geometric series sum in our sophomore math major Foundations of Mathematics course.
The students compare / contrast three or more proofs. In particular, they should consider these questions:
Does the proof apply to the finite geometric series or only the infinite series?
Where does the proof use |r| < 1? Is there any way to interpret the proof if r>1? What if -1<r<0?
Where is the limit concept implicit in the proof? In the classic proof, we use the fact that the limit of r^n as n goes to infinity. Is this implicit in this proof?
Which proof is most elegant? Most understandable?
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copyright: Besson and Styer, Villanova University 12/22/2007