Articles of infinity

What is infinity to the power zero

I have this notation: $$\lim_{k->\infty} k^ {1/k}$$ Is it correct to say that the output is 1, or is there some other result?

Residue at infinity (complex analysis)

I have trouble with the residue : at $z = \infty$. I tried to solve it at $z=0$ but it turns out that I was wrong while $z=0$ is not a pole. I must solve it at $z=2$ but I’m stuck. Any suggestion will be much appreciated.

I need to understand why the limit of $x\cdot \sin (1/x)$ as $x$ tends to infinity is 1

here’s the question, how can I solve this: $$\lim_{x \rightarrow \infty} x\sin (1/x) $$ Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is reformed in the following way $$\frac{\sin t}{t}$$ then, and this is what I really can´t understand, textbook suggest find the limit as $t\to0^+$ […]

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non–trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as non-branching trees of finite height: There are some crucial steps in the process of “understanding” larger ordinals, resp. when considering them as trees1. […]

random thought: are some infinite sets larger than other

This question already has an answer here: What Does it Really Mean to Have Different Kinds of Infinities? 9 answers

How to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$?

I need to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$.

Could you explain Perron's paradox to me, please?

This is Perron’s paradox: Let $N$ be the largest integer. If $N > 1$, then $N^2 > N$, contradicting the definition of $N$. Hence $N = 1$. What does it mean? I get from it that a very large number does not exist or $\infty=1$. Am I right? Or maybe the paradox is wrong?

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it possible that exist an infinite set $A$ with hartog number $h(A)=\aleph_0$?

Circle revolutions rolling around another circle

I just watched this video, and I’m a bit perplexed. Problem: The radius of Circle A is 1/3 the radius of Circle B. Circle A rolls around Circle B one trip back to its starting point. How many times will Circle A revolve in total? The intuitive answer is 3, but the correct answer is […]

Sum of infinite divergent series

I have learned that positive infinity plus negative infinity isn’t equal to zero, it’s an indeterminate form. However what happens if we subtract two infinite divergent series $\displaystyle{\sum_{n=1}^{\infty} n=\infty}$ ? Is it sill an indeterminate form or is it zero? $$\sum_{n=1}^{\infty} n-\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty} (n-n)=\sum_{n=1}^{\infty} 0=0.$$