July 16, 2025

Katelyn Tran

11th Grade

Fountain Valley High School

An Introduction to Series

Most people are familiar, to a certain degree, with sequences; at their core, sequences are an ordered list of numbers. Each element of a sequence is indexed by its particular position, and indices are generally natural numbers. So, some term a with index n in a sequence would be denoted as an, with the first term of a sequence given by a1, followed by a2, a3, and so on. It quickly becomes apparent in the application of both sequences and series that patterns arise; these patterns make sequences characterizable by rules that generalize any particular element in a sequence in terms of the index variable n. While some sequences have a finite number of terms, others continue infinitely following the pattern given by the general term.

A series, then, is the sum of all the terms in a sequence. While this is the basic description of a series, the greater significance lies in considering how the finite subsections of an infinite series constitute a sequence of partial sums. In other words, the partial sum Sn gives the sum of terms 1 through n but is also the general term in the sequence of sums {S1, S2, S3, … Sn} in a series with n terms. Even when series are infinite, they can still converge to a finite sum; in the context of partial sums, this means that each successive partial sum in the sequence approaches a particular number, and when n goes to infinity, Sn is a finite number.

The Power of Polynomials

While these principles alone are fascinating, the most incredible thing about series is the fact that they can be used to represent functions. For some function f(x), a series can form a polynomial capable of approximating its actual value, with the general term of an infinite series having two variables: an index variable and an input variable, which, in this case, is x. Let f(x) = ex. The example below is the polynomial approximation of the function with four terms:

The more terms the series has, the closer the polynomial estimates the function’s actual value, the same way adding more digits after pi yields a more accurate representation of its value. If the current pattern followed by the above polynomial is extended to infinitely many terms, the terms could be generalized as an = xnn!. The n is the index of the particular term, and x is the function’s input. The function’s value would then be approximated as Sn. Mathematicians have come up with similar series representations of numerous other functions as well, and it is from these principles that they have discovered series that converge to pi, an infinite number of them, as a matter of fact.

Pi and the Stars, and the Fundamental Forces of Physics

Indian astronomer and mathematician Madhava was credited for being the first individual to write series representations of trigonometric functions. His great interest in trigonometry is largely believed to be due to its relationship with his studies of astronomy. In the 15th century, Madhava discovered a series representation of the arctangent function. Using an input of 1, the series would yield a sum close to 4. With some basic manipulation, this series could then be used to represent pi. It would seem counterintuitive that pi, being a non-terminating and non-repeating (irrational) number, could be approximated by a polynomial; nonetheless, the achievement of such a polynomial adds yet another layer to pi’s complexity as a critical constant that appears regularly in mathematics. 

About two centuries later, this same series was independently rediscovered by mathematician James Gregory, and a few decades after this, Gottfried Wilhelm Leibniz discovered a specific case of it involving infinite limits as well. However, in August of 2024, a new revelation expanding on this series was uncovered by two Indian string theorists. Their intent was not the exploration of pi but the result of an attempt at uniting the fundamental forces of physics. While Madhava’s series gives pi as four times the quantity of the alternating series consisting of unit fractions of successive odd denominators, these physicists found that this series could be generalized to converge to pi with slight variations in accuracy and convergence rate depending on the input lambda, which can be any number. 

With the introduction of lambda into the general term and an infinite number of constants, it is now known that there exists an infinite number of distinct series that can represent pi; Madhava’s particular case was the result of allowing the limit of lambda to approach infinity. While accurate, the series converges very slowly, making the discovery of an infinite number of other series that could serve the same function more efficiently an exciting development, even in spite of the fact that other previously discovered series converge faster and still precisely.