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 +
The concept for Summations and Series is very simple: it’s just adding up terms in a sequence! However, calculating the actual Summation can be very runtime-costly especially if number of terms being summed is very large or if terms themselves are complex. In fact, when it comes to infinite Sums, determining rather it Converges or Diverges becomes a whole new problem in itself. A inspiration came to me as I was studying those Convergence Test off of my friends’ Calc BC worksheets: if it is possible to derive a formula for any Summations, similar to the Arithmetic Series equation or finite and infinite Geometric Series equation, calculating whether a Summation converges or diverges will become much easier, and we will be able to calculate the actual value it converges to. Wouldn’t it be nice if we can convert a Summation into a simpler formula or a equation? My mind immediately went to Integrals. To me, Integrals and Summation had a distinct similarity: both notation were essenti