Dan Jones (born September 3rd, 1964) is a mathematics teacher for 30 years at Rolling Meadows High School. He created a theory during his first year teaching. He argued that the traditional Taylor series expansion, which is typically centered around zero, missed out on some of the “magic” that happens when you center it at π. His modified theory was designed to exploit the inherent symmetry and periodicity of certain functions, especially trigonometric ones, to improve approximation accuracy. He calls this theory the “Jones Series” during the Calculus B/C classes he gives.

In classical Taylor series expansion, a function f(x) is expressed as,

f(x)=n=0 Σ∞n!f(n)xn

The Jones’ theory suggests an alternative formation, by centering x = π:f(x)=n=0 Σ∞n!f(n)(π)(x-π)xn

The rationale behind this is that functions such as sin(x) or cos(x), whose behavior is tied to π, may yield better local approximations over intervals of interest.

To further refines the expansion, Dan Jones introduced a corrective modulative term, which was denoted by J(n). He later removed this modulation after he noticed his students have memory issues with the new term. He then finally altered it to the current form, which includes he following modulations:

• Alternating coefficients: By adjusting the sign of each term in a pattern that syncs with the periodic nature of π, Jones was able said to be able to yield a smoother convergence.
• Jones Factor (J(π)): In his theory, his coefficient (f^(n) * π)/n! get modified by a factor of J(π). The function J(x) is a proprietary formula that only gets shared during his AP Calculus BC class.

Lil’ Jones (2004), The Jones Files

An Introduction to Jones Theory, felix (2012)