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The Taylor series is a method of turning non ‘nice’ polynomial functions into polynomial functions. By nice I mean: trigonometric, exponential, logarithmic. Now, this all probably sounds too good to be true and you may scratch your head in disbelief and ask yourself “How could it be possible to do such a thing?” And I say to that, sometimes dreams can come true.

0. A deep relation between derivatives and coefficients of a polynomial

Consider a general polynomial p(x) of degree m, then it must be in the form

Now, the question I propose is: how do the coefficients relate…


Following from the previous article I had written on the Taylor series here, in this article, I present a method for deriving the Leibniz product rule from Taylor’s theorem and Cauchy product rule.


The Leibniz rule gives us a nice closed form for the nth derivative of the product of two functions as a summation of binomial coefficients and derivatives of the individual functions:

The exponent denotes derivatives, so the ’n’ on the left side means the nth derivative of the product

Almost looks like a binomial theorem for expanding derivatives, doesn’t it? I wonder if there is a math field that studies sp00ky relationships such as these.

Cauchy Product Rule

The Cauchy product rule, in the context of this article…

In problem-solving,

There is a preconceived notion that mathematics is a subject for the gifted or very intelligent, that the whole idea is having hundreds of equations on a blackboard with symbols flying here and there. I’ll tell you here, that’s it not. When I think of mathematics, there are two parts for me: the idea and the execution.

The idea is the principle behind what I’m trying to accomplish and the execution is bashing the problem with compositions of known methods and tools.

When my study of mathematics began, I used to make a lot of mistakes in my calculations (…

Last time we calculated sums finite and infinite, applied calculus and counted using the geometric series. This time we will learn how we can apply derivatives onto the finite geometric series for evaluating other discrete sums. The main prerequisites of this article would be knowledge of the binomial theorem and basic differential calculus.

We start by considering a finite geometric progression of the form,


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