Some Intuition Readers who have gone through the previous sections may already suspect that the concept of a limit for functions should be analogous to the concept of a limit for sequences. Well those readers were not mistaken; they are indeed quite similar. Our main aim for this and next posts is to state limits and contuinity mathematically. However, I will show three eccentric functions to rebuild the contuinity intuition of readers. ...
I did not post anything for more than one month. I was having some problems. Maybe I lost my aim, maybe lost myself and maybe lost in myself. However, I have made up my mind to go on. I will continue to contribute and will ultimately finish Analysis I. Of course, I will then start to cover Analysis II. However, I am here and will be here for myself and most importantly for my small amount of readers. I put my anxiety and stress aside and cleared my mind. La mort du loup and then birth from ashes like a phoenix. Yes, it is what it is like to exist. Being in the world feels strange somehow, and writing digitally or conventionally is the cure to this. Writing anything. Any ink particule. In particular, writing with a fountain pen. It restricted me to write gibberish, to erase my mistakes and allowed me to think deeply, a little advise so to say. ...
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. ...