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Infinomial

The "infinomial" ("infino"-infinite "nomial- name) is a term first coined by Tim Baumgartner in the early 21st century during a Calculus lesson covering the rewriting of functions as infinite series. The exact conditions under which the name first came into origin is still disputed by scholars, as they debate over how much Baumgartner's most famous and brilliant student, Matthew Esparza, influenced his teacher with his supreme and undeniable awesomeness. With this said, several scholars attribute 50% of the labeling credit to what is now known, and widely accepted in the scientific community, as the Esparza Effect. The Effect is proven to have a substantially inspiring presence that can be recorded and quantified through simple experiment. The term's first recorded use was January 10th, 2012 at Redondo Union High School in Redondo Beach, California, in an Advanced Placement Calculus course.

The term "infinomial" originated as a variation of the more commonly known "polynomial" ("poly- many "nomial-name) which is a mathematical function that can be written as the addition or subtraction of a fixed number of expressions (ie 3 or 4); such as: y = x² + 2x + 4

The infinomial is the expression of a MacLaurin or Taylor Series, while applying the concept of a never ending number of terms.

The MacLaurin polynomial is a derived function that can be used to approximate the value of any given function at or close to a value "c". The MacLaurin polynomial can be written as

Mn(x) = f(c) + f'(c)(x-c) + f(c)(x-c)²/(2!) + f'(c)(x-c)³/(3!) + ... + fn(c)(x-c)^n/(n!) Where "c" is the focus point of the function f, and the point where Mn(x) gives an exact estimate of f(x). "n" is the number of factors taken. f' represents the first derivative of f; f, the second; f' the third and so on...

When the MacLaurin polynomial is turned into a series, the series is written as n=0 →∞ Σ [f^n(c)](x-c)^n/(n!) this equation is equal to the sum of every term of [f^n(c)](x-c)^n/(n!) between n=0 and n=∞ :

f(c) + f'(c)(x-c) + f(c)(x-c)²/(2!) + f'(c)(x-c)³/(3!) + ... + fn(c)(x-c)^n/(n!)

n is taken to be an infinite quantity, as opposed to the MacLaurin Polynomials which provide a designated, finite n value. This infinite polynomial will theoretically be equal to the exact function f(x).