Since the antiderivative of $\frac{1}{x}$ is $\ln(|x|)$, the surface under the graph of $\frac{1}{x}$ with $x>1$ is $\infty$. However, the antiderivative of $\frac{1}{x^2}$ is $-\frac{1}{x}$, so the surface under the graph of $\frac{1}{x^2}$ with $x>1$ is $1$. Although I understand how one can calculate this, I can’t really imagine it. Both graphs have an asymptote, […]

I know the line segment have a infinite number of points, but i know that exist different kinds of infinity ( $\aleph_0 $). My question is there same number of points on segment of line and entire line. If you know some good book or article about this topic i will be great full. Sorry […]

TL;DR: Meaning of these types of numbers: $1.2\overline{34}5\overline{67}$; There exist (rational) numbers that are non-terminating, but have a repeating form of digits (e.g., $ 1.2\overline{34} $). What I want to know is: does having 2 block of repeating digits make sense? My intuition says, “Why not?”; but my logic cannot comprehend, for example, when the […]

First of all I want to say that when coming to math – I know absolutely nothing – so please forgive me if my question is not “scientifically” correct, if it is not “syntax-correct” – or even too trivial or “stupid”. Is there a scientific name for 0.infinity ? (ZERO point INFINITY) And what would […]

What is cardinality of $P(\mathbb{R})$? And $P(P(\mathbb{R}))$? P is a Power Set, $\mathbb{R}$ is set of real numbers. In general – how can find cardinality of different sets? Is/are there a good method(s)? And is there some kind of handout, that provides list of cardinality of popular sets ($\mathbb{R}$, $\mathbb{Q}$, $P(\mathbb{Q})$) etc.? And, how to […]

There is a ball that starts at point A on a line and moves toward point B. Every second, it moves half of the distance left, but never stops moving: Etc. Would the ball ever reach point B? In one perspective, you could argue that since the ball is perpetually moving, it will reach the […]

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is the following. Clearly, ∞ isn’t a cardinal and $\aleph_0$ isn’t an extended real number, so we can’t […]

(I apologize if this is a duplicate, but I don’t know what terms to search for. Please feel free to close this if this has already been asked.) There are some properties of finite objects that don’t scale up to the infinite case. For example, any finite set of real numbers must have a least […]

Prove: $$\lim_{n\to\infty}r^n = +\infty\,, r > 1;$$ $$\lim_{n\to\infty}r^n = 0\,, 0 \le r < 1.$$ I am not quite sure how to prove this, but once someone proves it I will make sure to ask questions if I’m in doubt. Thank you very much! 🙂

This question already has an answer here: Why does $1+2+3+\cdots = -\frac{1}{12}$? 16 answers How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate] 4 answers

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