Infinity Definition and 988 Threads

Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol shown here.
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the Universe is infinite is an open question.

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  1. M

    Infinite Possibilities: Exploring the Different Types of Infinity in Mathematics

    How many kinds of infinity does math have? in my point of view "THERE ARE INFINITY KINDS OF INFINITY" What's your point?
  2. P

    Understanding Limits: Does it Exist or Go to Infinity?

    How can you tell if a limit does not exit or that it goes to infinity? examplelim\underbrace{x\rightarrow}_{0}(\frac{\sqrt{x+1}}{x}) The x goes to 0The book says the limit is \infty but if you take the left side limit you get -\infty and if you take the right side of the limit you \infty. So...
  3. R

    Why wave function should goto 0 f faster than 1/sqrt(x) at infinity

    Can anybody please explain the reason why a normalizable wave function ψ(x) → 0 faster than 1/√x as x → ∞. I can understand the reason why ∫ψψ*dx < ∞ But do not understand how quadratic integrability implies that. I would be very thankful to anybody who can give me some idea.
  4. R

    Zero to infinity can we assume same for universe ?

    Zero to infinity...can we assume same for universe ?? Hi, I am new to this forum, Basically I am not a scientist(not at all), like others in the forum, I belong to computer and Network field, However the science fantasies and attracts me very much just like others, ever since I was a kid , I...
  5. K

    Apostol: infinity as finite point

    I found a torrent online of Apostol's "Mathematical Analysis" 1st edition and I think I found a typo, or whoever scanned the book cut off the edge a bit... Apostol writes that the extended real number system R* is denoted by [-∞, +∞] while the regular real number system R is denoted by (-∞...
  6. P

    Multiply infinity by a positive number

    Homework Statement Prove that if limXn = +∞ and limYn>0 then limXnYn=+∞ The Attempt at a Solution limXnYn = limXnlimYn = (c)(+∞) where c is a positive real number I know in my head that a positive number multiplied by infinity is positive, but I am unsure how to prove this and we have...
  7. G

    Proof l'hopital for infinity over infinity

    i wonder about this proof for l'hopital for infinity over infinity: http://planetmath.org/encyclopedia/ProofOfLHopitalsRuleForInftyinftyForm.html how is this proved: http://bildr.no/view/1011658
  8. K

    Is a number preceding infinity, finite?

    Hi, I'm not sure if this is the right section, but I'm talking about numbers :). The questions is as written in the title: Is a number preceding infinity, finite?
  9. A

    Limit of sequence n^p/e^n as n approaches infinity

    Simple question just as the title says, but I can't remember or derive the solution for the life of me. I know that the answer is 0. I know why the answer is 0. But I need to know the mathematical derivation of the solution, and that's the part that I can't remember. So, to reiterate, how do you...
  10. O

    Limit to Infinity with Sqrt in Denom.(Is this correct?)

    Lim x→∞ \frac{7x^2-14x+7}{\sqrt{2x^4-4x^3+x+7}} Normally wouldn't have an issue here, just slightly confused by the sqrt. Attempted solution: \frac{7x^2-14x+7}{\sqrt{2x^4-4x^3+x+7}}*\frac{x^-2}{x^-2} Yields \frac{7}{\sqrt{2}} Is this correct? Similarly: lim...
  11. A

    What is the Infinity Norm & Why Use It?

    Hi I was wondering about the meaning of the infinity norm || x ||_\inf= max\{|x_1|, |x_2|...|x_n| \} if a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, why do we assign the maximum (or sup) as the value of this norm ? It must be a...
  12. I

    Exact value of infinity sum of Fourier series coefficients

    Homework Statement Sorry for doing another thread but I can't edit the old one any longer and I found out I made some calculation error but I'm pretty sure it's right now. The problem is to find the exact value of the series. Homework Equations \sum a_{k} The summation is to be done...
  13. M

    Understanding Limits at Infinity and Non-Existence of Derivatives

    Homework Statement Hello everyone, I am just new to this forum and also a beginner at calculus. I have a question from my textbook. It's: Find an example of f(x) that satisfies the following conditions : f(x) is differentiable for all x>0; limx->∞f(x) =2; limx->∞f'(x) does not exist...
  14. Y

    Can infinity fit in the palm of your hand

    I'm currently trying to reassess my image of the BB and the universe in general. I've been led to believe/understand that at the moment after the creation event the universe would fit millions of times within the space occupied by a single subatomic particle. However there's something wrong...
  15. J

    Limit at infinity, no vert asymptote, why?

    Homework Statement Find the vertical asymptote(n) and evaluate the limit as x \rightarrow n^-, x\rightarrow n^+, or state Does Not Exist. Homework Equations \frac{\sqrt{4x^2+2x+10}-4}{x-1} The Attempt at a Solution I have taken the limits at \pm\infty=2,-2 and understand those are my...
  16. M

    Proof that Bessel functions tend to zero when x approaches infinity

    I am aware that Bessel functions of any order p are zero in the limit where x approaches infinity. From the formula of Bessel functions, I can't see why this is. The formula is: J_p\left(x\right)=\sum_{n=0}^{\infty}...
  17. R

    How to Minimize the infinity norm of a matrix function

    Hi , I have been thinking of this question for a long time. Can someone give me an advice? There are three known matrices M, N, and K. M is a (4*4) matrix: M= [ 1 0 2 3; 2 1 3 5; 4 1 1 2; 0 3 4 3 ] N is a (4*3) matrix: N= [ 3 0 4; 1 5 2; 7 1 3; 2 2 1 ] K is a...
  18. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q.: The numbers \frac{1}{t}, \frac{1}{t - 1}, \frac{1}{t + 2} are the first, second and third terms of a geometric sequence. Find (i) the value of t, (ii) the sum to infinity of the series. Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a...
  19. Dotini

    Infinity at the Center of the Galaxy

    "Infinity" at the Center of the Galaxy http://www.sciencedaily.com/releases/2011/07/110719151234.htm http://www.wired.com/wiredscience/2011/07/milky-way-ribbon/ New observations from the Herschel Space Observatory show a bizarre, twisted ring of dense gas at the center of our Milky Way...
  20. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q.: A geometric series has first term 1 and common ratio \frac{1}{2}sin2\theta. Find the sum of the first 10 terms when \theta = \frac{\pi}{4}, giving your answer in the form h - \frac{1}{2^k}, where h, k \in N. Homework Equations Sn = \frac{a(1 - r^n)}{1 - r}, when...
  21. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q. Find the range of values of x for which the sum to infinity exists for each of these series: (i) 1 + \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ... (ii) \frac{1}{3} + \frac{2x}{9} + \frac{4x^2}{27} + \frac{8x^3}{81} + ... Homework Equations S\infty =...
  22. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q. Find, in terms of x, the sum to infinity of the series... 1 + (\frac{2x}{x + 1}) + (\frac{2x}{x + 1})^2 + ... Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a Solution S\infty = \frac{a}{1 - r} a = 1 r = U2/ U1 = (\frac{2x}{x + 1})/ 1...
  23. U

    What is the sum of n^2/n from 1 to infinity?

    Homework Statement \sum_1^\infty \frac{n^2}{n!} = The Attempt at a Solution Context: practice Math GRE question I don't know how to answer. Well, it's bigger than e and converges by the ratio test. Adding up the first 5 or 6 terms suggests that it converges to 2e. That's good enough for a...
  24. O

    Sum to Infinity of a Geometric Series problem

    Homework Statement Q.: A geometric series has first term a and common ratio r. Its sum to infinity is 12. The sum to infinity of the squares of the terms of this geometric series is 48. Find the values of a and r. Ans.: From textbook: a = 6, r = 1/ 2 Homework Equations...
  25. S

    Integral xsinx limits 0 to infinity

    Homework Statement So I'm trying to evaluate the following integral: 4\pi r^2{\int_0}^\infty r^2\frac{\sin{sr}}{sr}dr which after canceling out one of the r's, gives an integral similar to that of xsinx. I need to show that this integral vanishes for all values of s that are not 0...
  26. Z

    Show that x(t) approaches infinity in finite time

    Homework Statement Consider the equation \dot{x} = rx + x^3, where r>0 is fixed. Show that x(t) \rightarrow \pm \infty in finite time, starting from any initial condition x_{0} \neq 0. Homework Equations I can think of none. The Attempt at a Solution The idea alone of x(t) approaching...
  27. B

    The odds are one in an infinity

    If the odds of getting the right answer are one in an infinity, is it possible to stumble on to the right answer?
  28. C

    Proving "f(n) & s Have Same Sign as n Approaches Infinity

    Homework Statement if lim f(n) = s , prove that lim (f(n))^{1/3} = s^{1/3}. How do you know that as n approaches infinity, f(n) and s have the same sign. n is just an index in this case and f is not a function but a sequence. The attempt at a solution so we know that |f(n) - s| < epsilon...
  29. G

    Tan(pi/2)=complex infinity why?

    Homework Statement Hi, I have just now come to the realization that tan(pi/2) is not infinity but complex infinity. I was wondering why and can't seem to find the answer. I was told all through high school that tan(pi/2)=infinity or undefined but not complex infinity. Homework Equations The...
  30. A

    Exploring Infinity: The Concept of Infinite Numbers between 0-1

    Is it true, that there is an infinite amount of numbers between 0-1? Think about it, what number comes after 0, and before 1? Whenever I try to think about it, my mind goes blank. Discuss.
  31. E

    What Is the Real Part of x Divided by the Factorial of -1?

    given (-1)! = \tilde{\infty} , which is complex infinity the real part of \tilde{\infty} is \overline{?} which is "a quantity whose magnitude cannot be determined" as stated in wolfram's site at http://functions.wolfram.com/Constants/ComplexInfinity/introductions/Symbols/ShowAll.html"...
  32. A

    Prove that x[1/x] = 0 as x goes to infinity

    Homework Statement Prove that lim(x->infinity):x[1/x] = 0 by epsilon-delta defintion. (WITHOUT USING THE SQUEEZE THEOREM) The Attempt at a Solution well, It's easy to prove that x[1/x] approaches 0 as x goes to infinity using the squeeze theorem, but the question is to prove that without...
  33. P

    Complex Substitution and Infinity in Quantum Mechanics Integrals

    Homework Statement In Griffiths' Introduction to Quantum Mechanics problem 2.22 as well as 6.7, I used substitution to complete an integral. The original integral had limits from negative infinity to positive infinity. For my substitution, I had a complex constant term added to the original...
  34. G

    For what range of numbers the iteration x = x^2 -2 does not go to infinity?

    This is a problem related to julia sets but its more of a mathematical problem so I posted it here. x= x^2 - 2 For what values the iteration does not go to infinity. I can't figure out how to calculate that. I tried calculating a nth term of this in terns of initial term but all in vain.
  35. QuarkCharmer

    How do you take the limit of a function approaching negative infinity?

    Homework Statement How would you take this limit? The function is: lim_{x\rightarrow -∞}\frac{\sqrt{9x^{6}-x}}{x^{3}+1} Homework Equations The Attempt at a Solution Uh, square root of negative infinity? Graphing tells me that this should go to -3. I just recalled that I can...
  36. A

    Prove that the limit of sin(sqrt(x+1))-sin(sqrt(x-1)) at infinity doesn't exist

    Homework Statement The problem is to prove that the limit of sin(sqrt(x+1)) - sin(sqrt(x-1)) when x goes to infinity doesn't exist. The Attempt at a Solution well, I converted sin(sqrt(x+1)) - sin(sqrt(x-1)) into the alternative form -2sin(sqrt(x+1)/2 -...
  37. A

    Prove that the limit of [x]+[-x] at infinity doesn't exist.

    Homework Statement The problem is to prove that the limit of [x]+[-x] at infinity does not exist. The Attempt at a Solution I used the argument that the function [x]+[-x] is equivalent to the function f such that it gives 0 for all integers and gives -1 otherwise. therefore because the...
  38. C

    Can all real numbers be written down with an infinite amount of time and people?

    Could I write down all the real numbers from zero to 1? I know this sounds crazy. But let's say I have a person for every number between 0 and 1 . and then I tell them to write down a number different from every one else. Suppose they have the largest infinite amount of time to do this...
  39. BloodyFrozen

    Limits to Infinity: Exponential Function

    If we have an exponential fuction, (for example) Limx->∞ e(x2+2x+1)/(x2-3) Would we first determine the limit of the "argument" (not sure if right word) of ex and then replace the "argument" with the limit and then evaluate it? So for the example above, The limit of...
  40. 1

    What is the algebraic approach to finding limits approaching infinity?

    I'm having a hard time learning from the textbook, I know I can do this if someone just outlines what my goal is here... and what I can interpret from that goal. The solutions handbook just makes seemingly random algebraic changes to the limit function and then tells me what the answer is...
  41. O

    Does E(e^{-X}) = 0 imply X = \infty almost surely for X \geq 0?

    Does the following make sense: E(e^{-X}) = 0 \Rightarrow X = \infty\quad a.s. ? (Intuitively yes, but mathematically?) Thank you in advance for your help! :-) /O
  42. E

    Something weird about limit at infinity?

    Homework Statement This is a problem I came up with when I was doing something similar in Spivak's Calculus; although a simpler version. Suppose, we have f(x)=x^3 and g(x)=x^2 find \lim_{x\rightarrow \infty} f(x)/g(x) Homework Equations N/A The Attempt at a Solution...
  43. I

    How Can a Buffered Band Pass Filter Be Designed with an Infinite Resistive Load?

    Hello, I have a question. How do you design a buffered band pass filter with a resistive load of infinity? I have a feeling that a resistor with a resistance of infinity is an open circuit. Although how can this be implemented in a type of filter seen in the attachment? Thank you.
  44. F

    Minimizing infinity norm squared

    I have to minimize an expression of the following type: min <a,x>-L||x-u||_inf^2 s.t.: ||x||_inf <= R, where a is a vector of coefficients, x is the vector of decision variables, <.,.> denotes the scalar product, R and L are scalars, u is some constant (known) vector, and 'inf' denotes...
  45. N

    What is the difference between singularity and infinity in QTF theo?

    Please teach me this: What is the difference between singularity and infinity points.Because we often encounter with infinity counterterms in QTF theory,but trying to avoid the singularity counterterms. Thank you very much in advanced.
  46. A

    Limit of derivative as x goes to infinity

    Homework Statement Suppose that f and f' are continuous functions on \mathbb{R}, and that \displaystyle\lim_{x\to\infty}f(x) and \displaystyle\lim_{x\to\infty}f'(x) exist. Show that \displaystyle\lim_{x\to\infty}f'(x) = 0. Homework Equations Definition of derivative: f'(x) =...
  47. F

    Odd fraction includes pi and infinity

    does pi over infinity equal .00000...314159...? if not please post opinions. :(
  48. F

    Solving the Limit of a[n] as n Goes to Infinity

    Homework Statement [PLAIN]http://admitere.ncit.pub.ro/moodle/filter/tex/pix.php/5c2cef253f2db3240db03f8c9b6c9463.gif lim n->infinity of a[n] = ? Homework Equations |x| > 1The Attempt at a Solution Well, actually i figured out that the sequence converges, and I've tried to solve it using...
  49. narrator

    Exploring U: Can We Return to Earth Travelling at Light Speed?

    If we headed directly into space traveling at many times the speed of light (ignoring for a moment that you can't travel that fast), maintaining exactly the same course for the whole trip, would or could we eventually find ourselves heading back to Earth? I got a reply elsewhere, suggesting I...
  50. K

    Is Our Chemical Knowledge Truly Infinite?

    I'm interested in philosophers' opinions on a question that I posed in another forum (https://www.physicsforums.com/showthread.php?t=496119). As I say in my latest post, the entire thread was sparked by the comments of the Philosopher of Chemistry, Joachim Schummer: We have no reason at...
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