Sungjai's Recreational Mathematics

Summations are fun, but they can be really tricky. Most often, the fun properties come from the complexity of  terms defined within Sigma. What I became curious about is the properties of Sigma Summation itself: what interesting properties will Summation have when they are multiplied to each other, or even nested inside one other? That is the question we are going to explore with this article: tricks and properties of Summation Multiplication and Nested Summations! Before exploring our own properties, let us cover some of the more well known ones: $$\sum_{i=n}^m f(i) + \sum_{i=n}^mg(i) = \sum_{i=n}^m(f(i)+g(i))\\ \sum_{i=n}^mcf(i) = c\sum_{i=n}^mf(i)\\ \sum_{i=n}^m1 = m+1-n\\ \sum_{i=n}^mf(i) = \sum_{i=k}^mf(i) + \sum_{i=n}^{k-1}f(i)$$ With these simpler properties out of the way, let's begin with more exciting concepts: Multiplication $$\sum_{i=n}^mf(i) \times \sum_{j = a}^bg(j)$$ Let's start off simple: what comes after addition? Multiplication. The best way

In my last post I introduced my goals of $Ȣf(x)$, a transformation that produces a new function that fits the definition $\sum_{i=n}^mf(i) = \int_{n-1}^mȢf(x)dx$. In this article, we will explore few basic properties we can find of Ȣ Transformation without even needing to know how to transform works! Sum Rule of Essence  $$Ȣ(f(x)+g(x)) = Ȣf(x)+Ȣg(x)$$ This property becomes very obvious when we consider that Essence is related to Series. Let's consider $Ȣ(f(x)+g(x))$. By definition of Essence, we find that $$\sum_{i=n}^m(f(i)+g(i)) = \int_{n-1}^mȢ(f(x)+g(x))dx$$ We can modify the Summation on the left-hand side as such $$\sum_{i=n}^m(f(i)+g(i)) = \sum_{i=n}^mf(i) + \sum_{i=n}^mg(i)$$ Using Essence, we can represent each sub-summations as integrals $$\begin{align*}\sum_{i=n}^m(f(i)+g(i)) = & \sum_{i=n}^mf(i) + \sum_{i=n}^mg(i)\\ = & \int_{n-1}^mȢf(x)dx+\int_{n-1}^mȢg(x)dx\\ = & \int_{n-1}^mȢf(x)+Ȣg(x)dx \end{align*}$$ This leads to that $$\sum_{i=n}^m(f(i)+