Sequences

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. ...

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. ...

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? ...

Back then When I was in college, I used to solve a lot of sequence problems. I really liked the topic. It is something that I cannot depict right now, because I have only a few words in my mind. Whatever I write here cannot express the importance of the sequences in mathematics. Nevertheless, I will try to explain the topic while giving my understanding. While learning sequences it is important to build a bridge between the mathematics we know right know, and the mathematics that we will know in the future. ...