Articles of sequences and series

Excess Notation System

I know there are already threads that talk about Excess notation – but they don’t really clarify anything for me… What is Excess notation exactly (at a machine level)? And why, and how, would we ever use it? Here is the page from my textbook that is supposed to explain it: Excess System page from […]

A question about the closure of countable discrete subset

Let $X$ be Hausdorff. Given any countable discrete $D \subset X$, is the closure $cl(D)$ as a subspace of $X$ Frechet-Urysohn? Added: If I may ask more, is it must be discretely generated? A space is called discretely generated if for every $A\subset X$ with $x \in cl(A)$ there is a discrete $D\subset A$ such […]

Why not keep $\epsilon$ in Proof : Any Convergent Sequence is Bounded

I am trying to understand the proof of the proposition: Any convergent sequence is bounded. In my textbook, the author uses the definition of convergence for a sequence $\{a_n\}\to l$ and fixes $\epsilon=1$ so that there is a natural number $N$ such that \begin{align*}n>N&\implies|a_n-l|<1\\&\implies |a_n|<1+|l|.\end{align*} Then $\{a_n\}$ is bounded by $\pm U$ where $U=\max\{|a_1|,|a_2|,|a_3|,\dots,|a_{N-1}|,|a_N|,1+|l|\}$. What […]

Approximate a series with finite number of terms

How it is possible to approximate: $$\sum_{i=1}^{NR}{i\cdot \left( \dfrac{1}{1-p} \right)^i} $$

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an “ordered and infinite list of real numbers”. “Ordered” here means there is a definitive first, second, …, $n$ th term in the list. My initial reaction […]

Convergence of $\sum_{n=1}^\infty\frac{1}{2\cdot n}$

It is possible to deduce the value of the following (in my opinion) converging infinite series? If yes, then what is it? $$\sum_{n=1}^\infty\frac{1}{2\cdot n}$$ where n is an integer. Sorry if the notation is a bit off, I hope youse get the idea.

The definition of a sequence converging.

The sequence $\{a_n\}$ converges to t he number L if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$, $$n > N \implies |a_n – L| < \epsilon$$ If no such number $L$ exists, we say that $\{a_n\}$ diverges. If $\{a_n\}$ converges to L, we write $$\lim_{n\to\infty}a_n=L$$ I am […]

Convergence of $\sum_{n=1}^{\infty} \frac{\ln n }{\sqrt{n^3-n+1}}$

Why the following series $$\sum_{n=1}^{\infty} \frac{\ln n }{\sqrt{n^3-n+1}}$$ converges?

Showing that the pmf of a complicated expression sums to 1 (i.e. it converges)

Background For the past week, I have struggled with a complicated probability distribution in Mathematica. I want to analytically show that it is normalized, or at least, converges. This is the last part of a project in which I have finished all numerical verifications. I have rewritten the probability distribution as nicely as I could. […]

Creating an alternating sequence of positive and negative numbers

TL; DR -> How does one create a series where at an arbitrary $nth$ term, the number will become negative. I’m learning a lot of mathematics again, primarily because there are such wonderful resources available on the internet to learn. On this journey, I’ve stumbled across some very interesting sequences, for example: $$ a_n = […]