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In this post we will continue to investigate more theorems of convergence tests. Tests D’Alembert Ratio Test $\textbf{\small Theorem 1.17.:} $ Let $\sum_{n=1}^{\infty} a_n$ be positive series and let $l = \lim_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}$. Then, series converges if $l < 1$, series diverges if $l > 1$, and gives no information if $l = 1$ or limit fails to exist. $\textbf{\small Proof:} $ Let $l < 1$. We want to prove that $\sum_{n=1}^{\infty} a_n$ converges. Let us write what we know from assumption. Choose $\varepsilon > 0$ and $\forall n > N$ and $0 < (l + \varepsilon) < 1$: ...

Telescoping Series In this post, we will learn more about infinite series. We shall start with so-called telescoping series, that has a form of $$ \begin{aligned} \sum_{k=1}^{\infty} (a_n - a_{n+1}) \end{aligned} $$ $$ \begin{aligned} \sum_{n=1}^{\infty} (a_n - a_{n+1}) \end{aligned} $$ Let us investigate this sum step by step. Observe that the partial sum $s_n$ can be written as $$ \begin{aligned} s_n = a_1 - a_2 + a_2 - a_3 + \dots + a_n - a_{n+1} \end{aligned} $$ $$ \begin{aligned} s_n = a_1 - a_2 + a_2 - a_3 + \dots + a_n - a_{n+1} \end{aligned} $$ So we can just simplify this ...

By the intuition from our birth, we love summing things, categorising the similar things and so on. Consider a sum $A=1+2+3+4+\dots$ what we have here is infinite sum, as “$\dots$” imply “this sum goes to infinity with the order you see”. So in a some sense, it behaves like sequences, because we write the things in order like $1$, $2$, $3$, $4$, $\dots$ and then we sum them. Mathematicians tought that so, and as a result they came up with a new topic called “series”, especially “infinite series”. Spoiler alert, an infinite series can converge, if not we say series is divergent. ...

First Initiation When we first started to learn mathematics, our concern is only to solve problems, understand what equation means, comprehend what teacher says during class and etc. At high school we learn more about sets and functions. However, if we are not curious enough, then the philosophical meaning of the word “mathematical space” is not emphasized. When $f: \mathbb{R} \rightarrow \mathbb{Q}$ is written, everybody who has learned high school math can say “the function takes values from $\mathbb{R}$ and maps it to $\mathbb{Q}$. But have people reading this ever thought about the background? ...

Definitions As we discussed earlier in here differential equations widely used when we need to model real-world actions. For instance, velocity, acceleration, heat on a surface etc. Remember that, derivatives measure how a quantity changes, so a differential equation describes a relationship between a quantity and its rate of change. Let us leave these intuitive definitions aside and focus on rigorous definitions of differential equations. $\textbf{\small Definition 1.1.:} $ (Differential Equations) An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables is said to be a differential equation, shortly DE. ...

The converse of Theorem 1.2. (every convergent sequence is bounded) is not generally true. For example, observe the sequence $\left((-1)^n\right)_n$. The elements of these sequence consist only of $1$ and $-1$. It is bounded, yet not convergent. So we have a very great example of disproving a theorem by a counterexample. I love such examples because it is not important how big the theorem is, if you can find any counterexample then you can either throw that theorem into thrash bin or you can refine and redefine it. What we will do right now is refining and redefining. To do so, we need definition for monotonicity. ...

Why to use differential equations I do not and will not like giving plain formula or technic to solve problem, rather I want to learn the truth as Faust did. Faust was extremely successful person, he was some kind of researcher, he read and read. However, he was not satisfied with the position of his life. It seemed meaningles and pointless to him. One day he made a pact with Mephistoteles. The deal was pretty straightforward: in exchange for his soul he will gain unlimited knowledge and hedonic pleasures. If Faust had been a real person, he would have hated plain formulas given without reasoning. Therefore, we will not use any formula, theorem or lemma without proving it beforehand. I have briefly discussed Faust here. ...

Where does crisis come from In the last chapter we talked about Russel’s paradox and the need for a formalized system of mathematics. In this section of the blog, we will discuss why mathematics needed to be formalized and what ideas were involved. A mathematician called Cantor created the set theory in the 1870s, he was confident about his theory. He did make a real leap in mathematics in terms of modernization. I mentioned naive set theory a little bit here. Well, I’m not someone who can criticize Cantor, but Bertrand Russell did. He developed a very clever paradox that created a major bottleneck in set theory. ...

Intuition is a dangerous weapon Nearly every mathematician had very sharp and concise, yet sometimes dangerous intuition, which led mathematics into the very flawy statements. The theorems and even axioms in a system should be precise internally. I had talked about that topic in the page Gödel’s Proof. The definition (Definition 1.14) in section Analysis I Part 4 stated after the concept of convergence had been used without proof, relying only on intuition. Hopefully, it did not lead mathematics to flawy theorems, statements and etc. Why do we need such a solid definition of convergence? Well, it is because we need to prove them in general terms. Let us continue with boundedness and then move to the algebraic properties of limit? ...

Introduction In 1931 a mathematician with round glasses wrote a quite short paper with title “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme”. He was 25 years old. So young was he that, instead of tearing mathematics apart with the energy of his youth, he could have spent his time doing nothing. He was at the University of Vienna and since 1938 a permanent member of the Institute for Advanced Study at Princeton. ...