# Aziz Contractor Takes an “Infinite” Challenge… and Solves it with Help from a 19th Century Russian Mathematician

Clayton State University faculty members are not afraid to challenge their students, sometimes in difficult venues, as a means of empowering students to excel, both inside and outside the classroom.

One such professor is Associate Professor of Mathematics Dr. Elliot Krop, and one such student is double major (Math and Computer Science) Aziz Contractor.

Contractor, a native of India who came to the U.S. 10 years ago, was a student in Krop’s Calculus 2 class this past summer. In that class, Krop posed what he refers to as a “challenge problem” about the convergence/divergence of a special infinite series.

“In the latter half of Calculus 2, we learned about infinite sums,” explains Contractor. “These are just sums of sequences that increase or decrease in a specific pattern. Think of a sequence such as 1/k, where k starts at one and keeps growing. A sum of this sequence would simply add the consecutive terms.”

Sounds simple enough, doesn’t it? Well, Krop’s challenge was anything but simple, and Contractor had to learn some advanced calculus tests, including the little-known Ermakov's Test, developed by a Russian mathematician in 1871.

“We had to find a pattern which showed us how many logs we could take of a specific k value,” says Contractor, warming to the subject of advanced calculus. “This alone was a tedious task. Second, this was an infinite product inside an infinite sum and it was the first time I looked at something like this, let alone trying to solve it.

“I tried many tests and methods that I was familiar with but none of the tests and methods we learned in Calculus 2 were meant to deal with such a complex infinite series. Then, we came across Ermakov's Test. It tests for convergence and divergence and is meant to be used on infinite series that contain logarithms.”

“In order to solve the series he had to learn a few more advanced tests as well as proof techniques from analysis to understand why those tests are true,” adds Krop. “Eventually, he solved the problem by using Ermakov's Test.”

Such was the success of Contractor’s solution that he gave talks at two mathematics conferences on his result; the Kennesaw Mountain Undergraduate Mathematics Conference, and the 10^{th} Annual University of North Carolina Greensboro Regional Mathematics and Statistics Conference.

Contractor’s solution became even more impressive to Krop when he discovered a different proof to the problem, published by a mathematician at Cornell University almost 70 years ago, in 1947.

“I think that a Clayton State student independently proving a result published by a mathematician from Cornell is a notable achievement,” states Krop.

“We had to evaluate our sum over certain intervals of k in order to make sense of the results,” is Contractor’s explanation of his outcome. “For each interval for the values of k, we obtained a result that was close or equal to one when using Ermakov's Test. We were thus able to establish a pattern and conclude that this infinite sum diverges.”