Previously, we have explored methods to compute the essence of power functions $Ȣx^n$ which involves solving a large system of linear equations. This method is equivalent to solving for the inverse of a large $n\times n$ matrix where entries are values of pascal's triangle. Though the matrix method allow us to solve for large number of essences at once, it does not extend easily to solve for next iterations of essence coefficients. Rather than reusing the values we have already solved for, we will have to solve for inverse of a separate larger matrix again. Here we will introduce an iterative method for solving for these coefficients. Chapter 0: Recap Let us first remind ourselves of the definition of essence. For a function $f(x)$, we want to find the transformation $Ȣf(x)$ such that we are able to 'smooth out' its series: $$\sum_{i=a}^b f(i) = \int_{a-1}^b Ȣf(x) dx$$ For example, we can solve for the following functions: $$\begin{align*}Ȣ1 &= 1 \\ Ȣx &= x +
When you first learn of Summation in Pre-Calc class, it is probably in build-up to Riemann Sum. You are taught the basics of Summation and simple manipulations, such a $\sum_nc*a_n = c*\sum_na_a$ for constant $c$, but you don't get to explore deeper into Summations and really play around with it. One thing that really intrigued me was why Summations only incremented by a unit. When we have a Sum like $\sum_{n=3}^{10}a_n$, it is equal to $a_3+a_4+a_5+\dots+a_9+a_{10}$, always incrementing by a unit( $1$ ). Why couldn't we Sum with interval that is not a unit? How could we have something like $a_3+a_{3.5}+a_4+a_{4.5}+a_5+a_{5.5}+\dots+a_{8.5}+a_9+a_{9.5}+a_{10}$, incrementing by a half instead of a full unit? Sets Of course you technically can use Sets and Set Builder Notation to create a specific Summation. $$\sum_{i \in S}i,S=\{\frac{x}{2}|x\in\mathbb{N_0}\} \, = \, 0+\frac{1}{2}+1+\frac{3}{2}+\dots$$ The above is Sum of all Rational Numbers incrementing by $\frac{1