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Questions tagged [infinitesimals]

For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).

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Can we have complete closure for the known six arithemtical operators?

Lets define an implementation of the integer numbers as ordered pairs of cardinals with zero, so we stipulate that: $\langle 0,x \rangle = +x$ $\langle x,0 \rangle = -x$ Where $x=|X|$ for some set $...
0
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2answers
68 views

Proving $\displaystyle{\lim _{n\to\infty}}\sqrt[n]{n} = 1$ given that $\inf \{\sqrt[n]{n}|n\in \mathbb{N}\} = 1$ [duplicate]

Here's my attempt at trying to prove it by definition: $$\exists \mathcal{E} > 0 | \exists N\in \mathbb{N}|\forall n > N:$$ $$|\sqrt[n]{n}-1|<\mathcal{E}$$ But from this point on, I'm not ...
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38 views

Determine that if $a_n \longrightarrow L$ then $\sqrt[n]{a_1\cdot…a_n}\longrightarrow L$ [duplicate]

I managed to prove that when $a_n \longrightarrow L$ then $\frac{n}{\frac{1}{a_1}+...+\frac{1}{a_n}} \longrightarrow L$, and $\frac{a_1+...+a_n}{n}\longrightarrow L$, and all these things scream the ...
3
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2answers
77 views

Calculating $\lim_{n\to\infty}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} +\cdots+\frac{\sin(2\sqrt n)}{n\sqrt n\cos\sqrt n}\right)$

Using the trigonometric identity of $\sin 2\alpha = 2\sin \alpha \cos \alpha$, I rewrote the expression to: $$\lim_{n\to\infty}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + \cdots+\frac{\sin(2\...
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4answers
115 views

Calculating $\displaystyle {\lim _{n\to\infty}}\frac{1+2+3+…+n-1}{n^2}$

My first attempt was using limit arithmetic, but it fails because one of the operands is infinite, so that didn't work. I then tried using the squeeze theorem: $$b_n = \frac{1+2+3+...+n-1}{n^2}$$ $$...
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2answers
55 views

Proving $\lim _{n\to\infty}\frac{1}{ a_n} \neq \alpha$ given $\lim _{n\to\infty}a_n = 0$ and…

Given: $$\displaystyle {\lim _{n\to\infty}}a_n = 0\\\alpha \in \mathbb{R}\\a_n \neq 0$$ I'm trying to show: $$\exists \mathcal{E} > 0| \exists N \in \mathbb{N}|\forall n > N:$$ $$\left| \frac{1}{...
4
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4answers
85 views

Proving $\displaystyle{\lim _ {n\to\infty}}\frac{6n^3+5n-1}{2n^3+2n+8} = 3$

I'm trying to show that $\exists \,\varepsilon >0\mid\forall n>N\in\mathbb{N}$ such that: $$\left|\frac{6n^3+5n-1}{2n^3+2n+8}-3\right| < \varepsilon$$ Let's take $\varepsilon = 1/2$: $$\left|...
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1answer
27 views

Does the inverse function theorem provide a path to interpretation of more general infinitesimal quotients?

Let us first bring up inverse function theorem : If $y=f(x)$ and if $f'(x)$ exists at some point $x=a$, then exists in some neighborhood of $(a,f(a))$ an inverse function $f^{-1}(x)$ which around ...
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2answers
69 views

Manipulating infinitesimals - what is wrong with this argument?

Now I know that $dy/dx$ is not a fraction etc., but if you look at the works of 17th -- early 19th century mathematicians you'll see they played fast & loose with their infinitesimals and arrived ...
3
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0answers
55 views

Infinitesimal Cellular Automaton

I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following: Let $(X,*)$...
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1answer
27 views

The area $A$ of metric elements $(dx)^2$

I define a metric: $$ (d s)^2=(d x)^2+(d y)^2 $$ As per the pythagorean theorem, all terms of the infinitesimal metric $(ds)^2$, $(dx)^2$ and $(dy)^2$ are (infinitesimal) areas. I am trying to ...
2
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2answers
61 views

confused with using $dx$ [duplicate]

I am studying differential equations right now and I am confused the way $dx$ is being used. When I learnt calculus I thought that $\frac{df}{dx}$ is just symbolic representation of derivative and we ...
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0answers
22 views

Ultraproduct of a Function versus Functions of Ultraproducts

Let $G$ be a group (or even a set for our purposes here) and consider functions from $G$ to $\mathbb{R}$. Now after choosing a non-principal ultrafilter of $\mathbb{N}$, we can construct ultrapowers $^...
4
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0answers
78 views

Arc length of a Polar curve as a Riemann sum

Suppose we have a curve in polar plane satisfying the equation $r=f(\theta)$ with $\theta\in[a,b].$ To find the area enclosed by this curve in this range of $\theta$ using Riemann integrals, we ...
2
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1answer
49 views

On infinitesimal generator when the volatility of Brownian motion is given as a function of time

Consider standard Brownian motion. $dS_t=\mu\space d_t+\sigma\space dB_t$ For the process, the operator are given below. $Af(x)=\mu \frac{df}{ds}+\frac{\sigma^2}{2}\frac{d^2f}{ds^2}+\frac{df}{dt}$ I ...
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0answers
46 views

Order of $f(x)=\frac{3}{-2\ln|x|-2\ln|x+3|}$ with respect to $x$

I have $f(x)=\frac{3}{-2\ln|x|-2\ln|x+3|}$ and have to determine the order of this infinitesimal with respect to $x$ for $x\to0$.Then it doesn't matter if we are approaching from left or right because ...
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1answer
39 views

Infinitesimal order with Taylor series

I'm here again with another exercise (after this one) on Taylor series: determine the infinitesimal order for $x\to 0$ of the function $$ f(x)=\sqrt{\frac{x^4+x^5}{x-\sin{x}}}\ . $$ The problem is ...
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1answer
56 views

Order of $h(x)=\frac{\ln^5(1+x\ln^{1\over8}(x))}{x^x-1}$

I have to determine the order of infinitesimal $h(x)=\frac{\ln^5(1+x\ln^{1\over8}(x))}{x^x-1}$ with respect to $x$ for $x\to0^+$ $\ln^5(1+x\ln^{1\over8}(x))\sim (x\ln^{1\over8}(x))=x^5\ln^{5\over8}(x)$...
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0answers
54 views

what do infinitesimals look like?

i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions (1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a ...
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2answers
39 views

Adding changes or derivatives between two points

I've playing with the limit definition of derivative and I've to somewhat confusing conclusions. To clarify, I'm from an Engineering background so I don't think that an instantaneous rate of change ...
4
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3answers
112 views

Have I discovered a new significance to a previously discovered constant?

I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged ...
2
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1answer
48 views

Calculate $\lim_{x\to 0}\left(\frac{3\sin(x)-3x \cos(x)}{x^3}\right)^\frac{1}{x}$

This is an exercise from today's exam and I think I did something wrong: Calculate the limit: $$\lim_{x\to 0}\left(\frac{3\sin(x)-3x\cos(x)}{x^3}\right)^{1/x}.$$ To tackle this problem I used ...
2
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1answer
41 views

What are the prime ideals in the ring $\mathbb{Z}[i](\epsilon)/(\epsilon^2) $?

Let $\mathbb{Z}[i,\epsilon] \simeq \mathbb{Z}[i](\epsilon)/(\epsilon^2)\simeq \mathbb{Z}[x,\epsilon]/(x^2 +1, \epsilon^2)$ be the Gaussian integers with an infinitesimal number $\epsilon^2 = 0$. What ...
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2answers
32 views

Finding the infinitesimal order of a function as $n \to \infty$

I have to find the infinitesimal order of a function $f(n)$ as $n \to \infty$, this is what I did: $$ \begin{split} f(n) &= \sqrt{n+1} - \sqrt{n} + \frac{1}n \\ &= \frac{n\sqrt{n+1} - n\...
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1answer
50 views

Notation Question for “Generalization of L’hopital’s Rule” PDF by V. V. Ivlev and I. A. Shilin

Within this brief text starting on page 2, the author(s) have a habit of employing a sort of "mixed" notation of differentiation, mixing the "prime marks" of Lagrange's notation for differentiation of ...
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1answer
86 views

What is the product of infinitely many infinitesimals?

In general, taking the sequence for example, if $\lim\limits_{n \to \infty}a(n)=0$, we call the sequence $a(n)$ is an infinitesimal. It's well known that, the product of a finite number of ...
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3answers
53 views

When $x$ is given, what is $\mathrm{d}x$? [closed]

In the spherical coordinate, we know that $x = r\sin\theta\cos\phi$. Why does it imply that $$\mathrm{d}x = r\cos\theta\cos\phi \mathrm{d}\theta - r\sin \theta\sin \phi \mathrm{d} \phi?$$ It is first ...
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0answers
45 views

Two infinitesimals infinitely close to each other compared to a third one?

On keisler's book, Elementary Calculus, he presented the following definition: $\epsilon\ \approx\ \delta$ compared to $\Delta x$ if $\epsilon/ \Delta x \approx\ \delta\ / \Delta x$ and used it to ...
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2answers
3k views

Can we make it equal to x?

$\sqrt{6 +\sqrt{6 +\sqrt{6 + \ldots}}}$. This is the famous question. I have to calculate it's value. I found somewhere to the solution to be putting this number equal to a variable $x$. That is, $\...
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0answers
34 views

$x^2+y^2=0$ implies $x^2$ for every $x,y\in R$ is false

In Exercise 1.9 (iii) in A primer of infinitesimal analysis by John Bell we are asked to show that the following assertion is false: "$x^2+y^2=0$ implies $x^2=0$ for every $x,y\in R$", where $R$ is ...
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1answer
93 views

Proof that “$\uparrow$ is the unique solution of $tiny(G) = G$”

Tiny & miny games can be defined as: $$tiny(G) = \{0||0|-G\}$$ $$miny(G) = -tiny(G) = \{G|0||0\}$$ From the Wikipedia page for tiny and miny: Similarly curious, mathematician John Horton ...
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3answers
51 views

How to guess the output of $F(x)$ for $x$ that causes $0/0$

As an example: $$ F(x) = \frac{(1-x)^2-1}{(1+x)^2-1} $$ When $x=0$ this involves $0/0$ and calculators output 'undefined'. But when looking at a graph of the function it is intuitively clear that ...
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2answers
87 views

If $f'(x)=0$, is then $f(x+dx)=f(x)$?

I am always struggling with infinitesimals, and not sure I'm getting this right. The title basically states the simplest version of my question: If a function has zero slope at some point, is it ...
2
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1answer
119 views

The value of the infinitesimal in integral doesn't matter?

I am studying calculus by the infinitesimal approach using "Elementary Calculus: An Infinitesimal Approach" textbook. in page 187, the author proved that the value of the infinitesimal we integrate ...
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2answers
35 views

Validity of allowing $\epsilon$ to vanish in Baby Rudin Theorem 3.10a

In Theorem 3.10a of Rudin's PMA, we prove that $$\text{diam } \bar E = \text{diam } E$$ by [fixing] $\epsilon>0$ and [choosing] $p \in \bar E, q \in \bar E$. By the definition of $\bar E$ there ...
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0answers
53 views

Find infinitesimal transformation and generator

We have asked to find the infinitesimal transformation and generator for the following group of transformation. $ T_a: \bar{x}=ax$ and $\bar{u}=\frac{1}{a} u$ Please tell me how to find it. Thanking ...
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7answers
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Paradox about the volume of a cylinder

Trying to apply Cavalieri's method of indivisibles to calculate the volume of a cylinder with radius $R$ and height $h$, I get the following paradoxical argument. A cylinder with radius $R$ and ...
0
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1answer
30 views

Partial derivative of Algebra on infinitestimal

I'm seeing an infinitestimal equation in thermaldynmaic: $dE=TdS-PdV$. However, whats so interesting is that it's partial deritative was $\displaystyle(\frac{\partial E}{\partial V})_T=T(\frac{\...
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1answer
52 views

Is there a proper term for the rational and irrational infinitesimals [closed]

There are a different infinities, such as the countable infinity of the rationals, and the uncountable infinity of the reals, Aleph null and Aleph one. The conceptual infinitesimals of 1/infinity ...
7
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2answers
102 views

How Euler arrived at power series for $a^x$ and $\ln x$

I'm reading a book that reproduces Euler's arguments, and I have a few questions about a few things he does. Below are parts of the argument: Let $a > 1$. Consider an "infinitely small quantity" $...
3
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1answer
53 views

Order and principal part of $f(x)=\, e^{x^2} \int_{x}^{+\infty}e^{-t^2}\,dt$, infinitesimal as $x\to+\infty$

Determine if $f(x)$ is an infinite or an infinitesimal as $x\to+\infty$, its order and the principal part $$f(x)=\, e^{x^2} \int_{x}^{+\infty}e^{-t^2}\,dt$$ I can write, and then, using L'Hôpital's ...
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2answers
80 views

Can $\mathrm{d}x$ be thought of as a derivative and differentiation or it's just a small change in $x$ and nothing more? [duplicate]

The $\mathrm{d}x$ appears on integrals. I saw conflicting views regarding it. People sometimes write it does have a connection to differentiation and derivatives. Does it or does it not?
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1answer
58 views

Maximizing the value of a two variable function along any curve

I read that, of all the points on an origin-centered circle in the x-y plane, the function $z=ax+by$ is maximum (or minimum) at the point where $\frac{x}{y}=\frac{a}{b}$ I think this is too specific. ...
2
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1answer
94 views

Limit/Infinitesimal Expression for Probability of a Continuous Random Variable at Any Single Value in Its Support

The following is a lecture slide from a machine learning class: I already have basic understanding of probability, including continuous random variables. And I'm familiar with the typical ...
0
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2answers
50 views

Doubling dx and dy

Suppose we have a function $f(x,y)$. Thus the total differential is $df=f_x dx + f_y dy$. Does doubling $dx$ and $dy$ double $df$. I want to multiply everything by $2$ but since we are dealing with ...
2
votes
1answer
83 views

How to explain a continuous curve to a layman

In trying to explain irrational numbers, I am now wondering how to explain continuous curves. (Related to limits and such). In discrete systems (topology, geometry, etc.), a curve makes sense because ...
0
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1answer
73 views

Equivalent infinitesimal for $\log(\cos(x))$

I recently came across a limit problem which replaces $\log(\cos(x))$ with an equivalent infinitesimal $\cos(x) - 1$. How do we prove that $\cos(x)-1$ is the equivalent infinitesimal for $\log(\cos(...
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vote
3answers
81 views

Given $f(x,y)=5+2x+4y+x^2+y^2+(x^2y^4)^\frac15$, show that it is differentiable at $(0,0)$.

I was given the function: $f(x,y)=5+2x+4y+x^2+y^2+(x^2y^4)^\frac15$ I need to show it is differentiable at $(0,0)$. I started using the method of differentials and infinitesimal functions: $\...
1
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0answers
36 views

A question about nonparametric (kernel function) estimation

Suppose $X_1,\cdots,X_n\overset{iid}\sim F$, the $q$th quantile $\theta=F^{-1}(q)$. Let $K$ denote an $r$th order kernel. That is for some integer $r\ge2$ and constant $C\neq0$, $K$ satisfies $$$$ \...
0
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3answers
228 views

Are infinitesimals, i.e. $dx = …$, rigorous and correct notation? [closed]

In many fields of physics and engineering, when we want to describe an infinitesimal, for example, the electric field, we could say $dE(x_i,y_i) = e^{jkr}...dx_0dy_0$ Since derivatives are not ...