# Some student research projects

*An Unusual Application of Newton's Method* with Jennifer Sirkin

Jennifer and I proposed a new proof of divergence of the harmonic series. The original idea was to use Newton's method to generate a divergent sequence whose terms get closer and closer to each other (much like the harmonic numbers). We believe that our sequence is a kind of optimal example of one whose terms are greater than the harmonic numbers, but whose differences tend to zero. Our work is still in progress, but some of that work resulted in a new sequence in the OEIS. (Jennifer won 3rd place in the 2021 Skyway STEM competition.).

*Modeling the Boat/Bobber Problem * with Gabe McCandless, Tyler Metzger, and Alan Villalba

There is a common related-rates problem in which a boat is pulled toward a dock by a winch above the water level. The assumption is made that the boat will be pulled along the water until it hits the dock. This assumption is physically unrealistic because it results in the boat moving infinitely fast. We determined the physically correct model for the problem situation and confirmed the analysis with actual experiments. More details are available upon request and will (or may) be posted later.

*Partial Sums of the Prime Reciprocals Ordered with ln(ln (k+1))* with Cahron Cross, Jared Morton, and Jorge Sanchez

It is well known that the sum of the reciprocals of the prime numbers diverges. We are investigating the integer sequence whose nth term is the position of the nth partial sum of the prime reciprocals when jointly ordered with the values of ln(ln(k+1)) (starting with k=1). This sequence begins 5,10,18,28,38,51,64,79,95,.... More details will (or may) be posted later.

*A Closer Look at Bobo's Sequence* with Dan Clancy

Dan and I studied an integer sequence introduced by E. Ray Bobo in the article A Sequence Related to the Harmonic Series. Bobo posed several questions concerning the sequence; Dan and I answered them. Our research resulted in this paper. See sequences A103762 and A242679 in The On-Line Encyclopedia of Integer Sequences. (Dan won a 1st place in the 2011 Skyway STEM competition.)

*Exceptional Bobo Numbers* with Alan Mitchell and Sam Zandi

In the work with Dan Clancy described above, Dan and I found that some of the numbers in the sequence of Bobo numbers had an rather unusual property. These unusual Bobo numbers were called "exceptional," and we knew of only one exceptional Bobo number (namely, 36). Alan, Sam, and I took up the task of finding and describing all exceptional Bobo numbers. Our research resulted in a paper that has been submitted for publication (details available upon request). See sequence A277603 in The On-Line Encyclopedia of Integer Sequences.

*Kimberling's Conjecture on Harmonic Numbers when Jointly Ordered with Logarithms* with Gary Rathbone and Andrew Severs

Gary, Andrew, and I studied the sequence A206911 in The On-Line Encyclopedia of Integer Sequences. We have proven Kimberling's conjectures associated with the sequence and corrected the posted sequence. More details are coming soon. (This project won a 2nd place in the 2017 Skyway STEM competition.)

*The Falling Ladder Paradox Revisited* with Brittany Burke and Zach Jackson

The falling later problem is a famous related-rates problem that typically appears in Calculus 1. The standard related-rates approach to solving the problem leads to a physically unrealistic solution. Physically correct solutions are well known (but not in calculus classes). Brittany, Zach, and I actually performed some falling ladder experiments, analyzed the solution model, and provided some new details. Our work resulted in this paper (Preprint available here.)

*Lovie Smith, NFL Coaches, and Separating Hyperplanes* with Timothy Dannels, Nandjui Koutouan, and Reginald Thompson

We found a hyperplane in 7-dimensional space that best separated "winning" and "losing" NFL coaches. Lovie Smith falls on the losing side of our hyperplane. (This project won a 1st place in the 2013 Skyway STEM competition.)

*Generating cycles of Newton's Method* with Kevin Kuipers

Kevin studied the behavior of certain periodic cycles generated when Newton's method is applied to polynomials.

*Last updated January 20, 2023.*