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: $$\begin{align*}Ȣ1 &= 1 \\ Ȣx &= x +...
The concept for Summations and Series is very simple: it’s just adding up terms in a sequence! However, calculating the actual Summation can be very runtime-costly especially if number of terms being summed is very large or if terms themselves are complex. In fact, when it comes to infinite Sums, determining rather it Converges or Diverges becomes a whole new problem in itself.
A inspiration came to me as I was studying those Convergence Test off of my friends’ Calc BC worksheets: if it is possible to derive a formula for any Summations, similar to the Arithmetic Series equation or finite and infinite Geometric Series equation, calculating whether a Summation converges or diverges will become much easier, and we will be able to calculate the actual value it converges to.
Wouldn’t it be nice if we can convert a Summation into a simpler formula or a equation?
My mind immediately went to Integrals. To me, Integrals and Summation had a distinct similarity: both notation were essentially adding up values within a certain bound( Summation added up consecutive terms in Sequence, Integral added up infinite lower dimensions within bounds to calculate higher dimensional value such as Area under a curve). In fact, we first define Integral by using Summations with Riemann Summation.
What if we could go the other way around? Define Summations using Integrals?
Between $0$ and $1$, area under that interval will represent the first term of sequence $a_1$; between $1$ and $2$, interval will have area equal to $a_2$, and so on.
$$a_0=\int_{-1}^0f(x)dx\\
a_1=\int_0^1f(x)dx\\
a_2=\int_1^2f(x)dx\\
\vdots$$
Between interval from $n-1$ to $n$, the Integral of the curve will equal to $a_n$, the $n$-th term of the sequence.
$$a_n=\int_{n-1}^nf(x)dx$$
Through this representation of Sequence, we will be able to convert Summation problem into simpler Integral problem.
Summing up consecutive terms of sequences becomes summing Integrals with consecutive bounds, which in turn becomes a single Integral of larger bound.
$$\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx\\
\therefore \sum_{i=n}^m a_i = \int_{n-1}^m f(x) dx$$
Arguably, computing a single Integral will be less costly and easier to calculate as bounds become larger and larger compared to Summing up each terms individually.
Now the problem only becomes what function correspond to the sequence in such a manner?
$$\sum_{i=n}^mf(i)=\int_{n-1}^mg(x)dx$$
What kind of transformations must we apply to $f(x)$ to get $g(x)$?
$$\sum_{i=n}^mf(i) = \int_{n-1}^mȢ(f(x))dx$$
The $Ȣ$ denotes the transformation of a function to yield another curve such that the unit integral of $Ȣf(x)$ represents $f(x)$.
So in a sense, $Ȣ$ acts similarly to $\frac{d}{dx}$ and indefinite $\int dx$.
The $Ȣ$ symbol itself looks similar to Taurus Zodiac symbol and on computer will be typed as the Latin Ou Ligature(Unicode U+A64A).
Wikipedia Latin Ou (Ligature)
$$Ȣf(x) = g(x)$$
I will call the $Ȣ$ symbol itself the Ou symbol and I will pronounce $Ȣf(x)$ as the Essence of $f(x)$, similar to “derivative of” or “integral of” as I believe the resulting curve is essentially linked to the initial function $f(x)$.
The resulting function $g(x)$ will be called the Essence or the Sequence Curve and will follow the property outlined above.
Of course, the names and symbols may change later on, but this is how I’ve used them in my own notes so far.
I have few goals I wish to achieve:
First, be able to compute the Essences for the major functions, such as Power Functions, Trigonometry Functions and even logarithmic and Fraction Functions.
Second, I want to find general Rules to compute each Essences, such as the general and simple rules of finding the Derivatives.
Lastly, I want to learn the Essence equivalencies of Chain Rule and Quotient Rules; how $Ȣf(x)$ and $Ȣg(x)$ will relate to $Ȣ(f(x)g(x))$.
Another benefit of using Sequence Curves to calculate Summations is that using this method, you will be able to perform Summations with non-integer bounds. Similar to how Gauss's Gamma Function interpolates through factorial, Integration of Essence of a Sequence will interpolate through integer Summation.
Some of these goals are completed or nearly done, others have barely begun. Maybe some are even impossible, but I still want to try explore deeper into this connection.
Maybe someone else had already done this hundreds of years ago, but I want to try discovering this myself. Even if this is a trivial and insignificant contribution to maths, I still want to be able to be able to claim that I had done this myself.
That is my ambition for this project and if the journey interests you in any way, come along down my path through Sequence Curve and Summation to Integrals on this blog.
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 essentially adding up values within a certain bound( Summation added up consecutive terms in Sequence, Integral added up infinite lower dimensions within bounds to calculate higher dimensional value such as Area under a curve). In fact, we first define Integral by using Summations with Riemann Summation.
What if we could go the other way around? Define Summations using Integrals?
CONCEPT
What I envisioned is a curve that will represent a Sequence. The area under that curve will represent each terms of that sequence.Between $0$ and $1$, area under that interval will represent the first term of sequence $a_1$; between $1$ and $2$, interval will have area equal to $a_2$, and so on.
$$a_0=\int_{-1}^0f(x)dx\\
a_1=\int_0^1f(x)dx\\
a_2=\int_1^2f(x)dx\\
\vdots$$
Between interval from $n-1$ to $n$, the Integral of the curve will equal to $a_n$, the $n$-th term of the sequence.
$$a_n=\int_{n-1}^nf(x)dx$$
 |
| Visualizing how Integral of the curve will represent each term in a sequence. |
Summing up consecutive terms of sequences becomes summing Integrals with consecutive bounds, which in turn becomes a single Integral of larger bound.
$$\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx\\
\therefore \sum_{i=n}^m a_i = \int_{n-1}^m f(x) dx$$
 |
| Visualizing Summation being represented as integral of Sequence Curve. |
Now the problem only becomes what function correspond to the sequence in such a manner?
$$\sum_{i=n}^mf(i)=\int_{n-1}^mg(x)dx$$
What kind of transformations must we apply to $f(x)$ to get $g(x)$?
Ȣ Notations
To answer the question above, I will be using the $Ȣ$ Notation, as such:$$\sum_{i=n}^mf(i) = \int_{n-1}^mȢ(f(x))dx$$
The $Ȣ$ denotes the transformation of a function to yield another curve such that the unit integral of $Ȣf(x)$ represents $f(x)$.
So in a sense, $Ȣ$ acts similarly to $\frac{d}{dx}$ and indefinite $\int dx$.
The $Ȣ$ symbol itself looks similar to Taurus Zodiac symbol and on computer will be typed as the Latin Ou Ligature(Unicode U+A64A).
Wikipedia Latin Ou (Ligature)
$$Ȣf(x) = g(x)$$
I will call the $Ȣ$ symbol itself the Ou symbol and I will pronounce $Ȣf(x)$ as the Essence of $f(x)$, similar to “derivative of” or “integral of” as I believe the resulting curve is essentially linked to the initial function $f(x)$.
The resulting function $g(x)$ will be called the Essence or the Sequence Curve and will follow the property outlined above.
Of course, the names and symbols may change later on, but this is how I’ve used them in my own notes so far.
Goals and Ambitions
I believe that once fully understood, concept of Sequence Curves an be a very useful tool.I have few goals I wish to achieve:
First, be able to compute the Essences for the major functions, such as Power Functions, Trigonometry Functions and even logarithmic and Fraction Functions.
Second, I want to find general Rules to compute each Essences, such as the general and simple rules of finding the Derivatives.
Lastly, I want to learn the Essence equivalencies of Chain Rule and Quotient Rules; how $Ȣf(x)$ and $Ȣg(x)$ will relate to $Ȣ(f(x)g(x))$.
Another benefit of using Sequence Curves to calculate Summations is that using this method, you will be able to perform Summations with non-integer bounds. Similar to how Gauss's Gamma Function interpolates through factorial, Integration of Essence of a Sequence will interpolate through integer Summation.
Some of these goals are completed or nearly done, others have barely begun. Maybe some are even impossible, but I still want to try explore deeper into this connection.
Maybe someone else had already done this hundreds of years ago, but I want to try discovering this myself. Even if this is a trivial and insignificant contribution to maths, I still want to be able to be able to claim that I had done this myself.
That is my ambition for this project and if the journey interests you in any way, come along down my path through Sequence Curve and Summation to Integrals on this blog.
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