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)+...

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...

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...

Vector is a useful tool in maths and in physics that describes both magnitude and direction in a space. Vectors are used to describe concepts where direction matter, such as forces and velocity. For example, two cars can have same speed(scalar) but if one if heading north and the other south, they will have different velocity(vector). Vectors are usually written in terms of their components. Components are magnitude of the vector in one dimension. In 2D space, vector can be represented as coordinate of 2 components, <x,y>, and in 3D space, vector can be represented as coordinate of 3 components, <x, y, z>. Breaking down vectors into their components is extremely useful, especially in physics, because it allows one to break down complicated changes in vector into simpler and easier-to-work-with changes in its components. Take a look at projectile motion, for example. Velocity of an object fired up into the sky at an angle changes both its direction and magnitude. Attem...