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. A

    Formal definition of limits as x approaches infinity used to prove a limit

    Homework Statement use the formal definition to show that lim as t goes to infinity of (1-2t-3t^2)/(3+4t+5t^2) = -3/5 Homework Equations given epsilon > 0, we want to find N such that if x>N then absolute value of ((1-2t-3t^2)/(3+4t+5t^2) + 3/5) < epsilon The Attempt at a Solution...
  2. O

    MHB How can I prove that the infinite union of certain closed sets is not closed?

    Hello everyone! I'm trying to find a set a closed set ${A_n}$ whose inifinte union is not closed. Now, I can picture the following: If I let $A_n=[-\frac{1}{n}, \frac{1}{n}]$, then $A_n$ is closed, but their union is not, simply because the point $x=0$ seems to be a limit point at...
  3. D

    Precise definition of limits at infinity

    Homework Statement Let f be a continuous function on ℝ. Suppose that \mathop {\lim }\limits_{x \to - \infty } f(x) = 0 and \mathop {\lim }\limits_{x \to \infty } f(x) = 0. Prove that there exists a number M > 0 such that \left| {f(x)} \right| \le M for all x \in ℝ. Homework Equations...
  4. A

    What are the different types of infinities?

    How can there be different types of infinities? I just learned that cardinality of set of natural numbers, integers, prime numbers is alepho. Why are we replacing infinity with a number i.e. alepho. And what further blows my mind is that cardinality of real numbers is also infinity but a...
  5. M

    Don't understand convergence as n approaches infinity

    Here's the deal... I don't understand the limit as n→∞ of [(1+(.05/n))^20n -1]/[.05/n] My Calculus book says that it's supposed to approach {e^[(.05)(20)]-1}/[.05], but the numerator is a constant while the denominator goes to 0 as n→∞. The textbook, by Dr. Gilbert Strang, has similar limits...
  6. J

    Improper Integral with Infinity in Limits

    ∞ ∫xe^[-x^2] dx -∞ So basically I've solved for everything in this problem and it looks like it should be an indeterminate form and thus divergent. My book and Wolfram both say it's 0 and convergent though. I get it down into: lim [[e^(-t^2)] - e^0]/2 + lim [e^0 - [e^(-v^2)]]/2...
  7. nukeman

    Simple Limits question regarding infinity

    Homework Statement I am doing some squeeze theorem questions, and I always run into things that is something divided by ∞, or something divided by √∞ Why is it always zero? like... -3/√∞ = 0 or -1/∞ = 0 why?? Homework Equations The Attempt at a Solution
  8. T

    Proving n = (x^n)(e^-x) integrated from 0 to infinity

    Homework Statement Prove \int_{0}^{\infty}x^{n}e^{-x} dx = n!Homework Equations 0! = 1 (by convention)The Attempt at a Solution Basic step: n=0 \\ \int_{0}^{\infty}x^{0}e^{-x} dx\ = 0! = 1\\ \int_{0}^{\infty}e^{-x} dx\ = -[e^{-\infty}-e^{0}]\\ -[e^{-\infty}-e^{0}] = -[\frac{1}{e^{\infty}}-1]\\...
  9. S

    What Is the Limit of (-1)^n/n as n Approaches Infinity?

    Homework Statement Find the limit as n tends to \infty of: \frac{(-1)^n}{n} Homework Equations The Attempt at a Solution I know that (-1)^n alternates between 1 and -1. I also know that the limit of 1/n is 0. But I don't know how to compute the above limit. Any ideas will...
  10. D

    Show that (q_n)^2 approaches 2 as n goes to infinity

    For every natural number n, associate with it a natural number mn such that \frac{{m_n}^2}{n^2} < 2 < \frac{{(m_n + 1)^2}}{n^2} . show that {q_n}^2 approaches 2 as n approaches infinity. i need to show that |\frac{{m_n}^2}{n^2} - 2| < ε . what i have so far is that 2 -...
  11. S

    Prove sequence diverges to infinity

    Homework Statement I have to prove that the sequence a(n)=(n^3-n +1)/(2n+4) diverges to infinity. Homework Equations The Attempt at a Solution Observe that n^3-n +1 > (1/2)n^3 and 2n+4≤4n in n≥2 I am now stuck on how to proceed. I am confused on opposite inequalities for...
  12. K

    What is Space outside of the universe and infinity according to Physics?

    Space: So far for me it's 3D along with Time T. say 3D+T=4D (spaceTime), from the bigbang we have considered Time (t sub I) as well, considering Time from the start of bigbang to 13.7 Billion years after the bigbang. Since our universe is accelerating due to Dark energy, expansion is speeding...
  13. J

    Gamma Function Limits: Proving \Gamma(x) \to 0 as x \to -\infty

    Homework Statement The absolute value of the gamma function \Gamma (x) that is defined on the negative real axis tends to zero as x \to - \infty . Right? But how do I prove it? Homework Equations The Attempt at a Solution I've tried to use Gauss's Formula...
  14. A

    At what distance does a charge attain half its speed at infinity?

    Homework Statement A charge of 3.40μC is held fixed at the origin. A second charge of 3.40μC is released from rest at the position (1.25m, 0.570m). If the mass of the second charge is 2.49 m, and its speed at infinity is 7.79 m/s, at what distance from the origin does the second charge attain...
  15. PrincePhoenix

    Solving mgh at Infinity: Understanding GPE & GMm/r

    According to the eq. U=mgh the gravitational potential energy should keep increasing with height. But it actually reaches zero at infinity. At what point does the g.p energy start decreasing when going away from earth? Why does this eq. seem to contradict -GMm/r where g.p energy decreases with...
  16. P

    Limit as x approaches infinity

    Homework Statement Use algebraic manipulations to evaluate the limit below. lim (x^2 +4)/(x+3) X->inf Homework Equations The Attempt at a Solution (x^2 +4)/(x+3) (1+4/x^2)/(1+3/x^2) 1/1=? I My first couple attempts i got wrong so I'm trying again. Am I right this time around?
  17. S

    What is the issue with an infinite universe on a N-sphere?

    There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?
  18. V

    Question about limit of wave function as x goes to infinity

    So I understand why the limit of the wave function as x goes to infinity is 0. But on pg 14 of Griffiths 2nd ed. qm for example, why does he call lim x\rightarrow\infty ψ*\frac{dψ}{dx} = 0? How can you assume that \frac{dψ}{dx} doesn't blow up at x = ∞
  19. bcrowell

    Define Black Holes w/o Asymptotic Flatness?

    I see statements that in order to define a black hole, we need asymptotic flatness, but this only seems to be necessary because we want to define the horizon as the boundary of a region from which light can't escape to null infinity (\mathscr{I}^+). It seems like you can have a well defined null...
  20. clope023

    Integrate (x^2)exp(-x^2)dx from - to + infinity

    Hello, I'm doing a quantum mechanics problem from Griffiths chapter 1 where I'm trying to find the expectation value of x^2 given the function ρ(x) = Aexp[-λ(x-μ)^2]. The problem is in the title, but I'll rewrite it here simplified: ∫(x^2)exp(-x^2)dx Subbing u = x^2 <=> x = sqrt(u) => dx...
  21. R

    Maxima of Two Waves Beating Together: 0 or Infinity?

    Two waves of equal intensities "I" beat together, what is the maxima of the beat equal to ? Is it 0 or infinity,or other answers... I just couldn't get it
  22. J

    Proof of that a limit -> 0 as n-> infinity

    Hey, I've been trying to work out the following question, This is it including what I hope is an ok, or on the way to an ok proof. http://img269.imageshack.us/img269/6156/proofxq.jpg The book has a hint given that states there exists N in naturals such that |z_n|<ε/2 for all n>N...
  23. J

    Question: are there proportions in infinity?

    Let me preface this by saying Math is not my strongest subject. I am curious if there is any relationship between proportionality and the term infinity. This question stems from an online discussion about whether or not the physical universe is infinite or not. (Obviously its not, but I...
  24. B

    Can you use induction on n cases (as opposed to infinity)?

    Homework Statement this is probably a dumb question, but I'm doing this proof where i have to show two sets are equal, where each set is a union from 1 to n sets. this is pretty easy to show with induction, i think, but I'm used to using induction when i have an infinite amount of things, so...
  25. J

    MHB Integral from negative infinity to infinity

    $\displaystyle (1)\;\; \int_{-\infty}^{\infty}\frac{x^2}{1+4x+3x^2-4x^3-2x^4+2x^5+x^6}dx$
  26. J

    Is probability meaningful in cases of infinity?

    Is it meaningful to speak of probability in cases of infinity? For instance, consider me having an infinite line of balls arranged in the manner: - Red, Green, Blue, Red, Green, Blue, Red... Now, I'm picking a ball randomly from this line. Am I allowed to ask the question, "What is the...
  27. J

    Sum to the infinity of a series

    Homework Statement to find the value of \sum over n=1 to ∞ of [1/{1+(n-1)2}](1/3)^{2+(n-1)2} Homework Equations The Attempt at a Solution I have tried to solve in the way the arithmetico-geometric series are solved and tried to bring it in the form of the expansion of ln(1+y)...
  28. U

    L ' Hopital's Rule for x -> infinity

    Homework Statement Hopital's Rule for x -> ∞ applies the same way as x -> 0. Homework Equations As shown in attached pic. The Attempt at a Solution I tried to prove that x->∞ works the same way as x -> 0, only to get its reciprocal.. Not sure what is wrong with my working..
  29. O

    Second ODE, initial conditions are zeros at infinity

    second ODE, initial conditions are zeros at infinity! I want to know the temperature profile of phase transition layer in the interstellar medium. For stationary solution, the dimensionless differential equation I ended up with is \frac{d^2T}{dx^2} = \frac{f(T)}{T^2} - \frac{1}{T} where f(T)...
  30. M

    Lim as n approaches infinity for n/2^n

    1. Determine whether the sequence converges or diverges. If it converges, find its limit. {\frac{n}{2n}}+∞n = 1 I know that we have to use L'Hopital's rule for this, because as n increases, both the numerator and the denomator approach infinity. However, doing that gives me...
  31. C

    Principal Quantum number: n to infinity?

    I am aware that n is the principal quantum number and determines the energy of a specific energy level of an atom. In my notes, I see that n goes from 1,2,3... which implies to me all the way to infinity. If this is the case, why doesn't this imply that there can be infinitely many shells in an...
  32. 1

    Behavior of DE as t approaches infinity

    Homework Statement y' = -2 + t - y Draw a slope field and determine behavior as t -> infinity Homework Equations The Attempt at a Solution This is the first DE in my book that includes t in the differential equation. The slope field looks pretty wacky. For the others I was...
  33. marellasunny

    What is the concept of 'one infinity' in complex analysis?

    'In the real case,we can distinguish between the limits (+infinity) and (-infinity),but in the complex case there is only one infinity'-this was given in Lars Ahlfors's book on complex analysis. Can someone explain what he means by 'one infinity'? My proffesor asked me to look into the concept...
  34. D

    Concept of a limit being infinity

    This is more of a conceptual question dealing with a homework problem than with the problem itself... So I am asked to find various limits for the function (x2-9x+8)/(x3-6x2). All well, no problems... I am asked to find the left and right hand limits as x→0, which is -∞ for each. At the end...
  35. C

    Integral of exponential distribution from zero to infinity

    Homework Statement here's a problem from my assignment let integral p(x)dx=Ae^-(x/a) dx...(1) find value of A, that makes integral p(x)dx=1; and find mean x so that integral x*p(x)dx=a...2) Homework Equations now to solve the first one i found out A to be (-1/a*e^(x/a)) but...
  36. N

    Integral Sin(ax)^2 Between Infinity and 0

    Need result for integral Sin(ax)*Sin(ax) Between Infinity and 0 Cant find this anywhere but there is a standard result with a in it.
  37. D

    Potential difference in a sphere relative to infinity

    Homework Statement A thin plastic spherical shell of radius a is rubbed all over with wool and gains a charge of -Q. What is the potential relative to infinity at location B, a distance a/3 from the centre of the sphere?Homework Equations \text{$\Delta $V}=\int...
  38. M

    Understanding the role of infinity into the Multiverse.

    Hi there, I have no mathamatics background or never attended ant classes of Cosmology. So please if you can answer this question please do so without any Math :D please bare with me, English is not my native tongue. This is my question: according to some scientist the (visible)...
  39. E

    An example of a continuous function in L1 space with no limit at infinity

    Homework Statement I am trying to come up with a continuous function in L1[0,infinity) that doesn't converge to 0 as the function goes out to infinity. Homework Equations I am trying to show an example of an f in L1[0,infinity) (i.e. ∫abs(f) < infinity) where the limit as the function...
  40. A

    Building infinity corrected microscope from non-infinity corrected objective

    Hi there As is known using non-infinity corrected objectives in infinity corrected microscopes is not a good idea because of many reasons (image quality will be degraded, back focal plane will be in wrong place, parfocal distance is not preserved, etc.). What about building infinity corrected...
  41. P

    Riemann Zeta approaches infinity as x approaches 1

    Homework Statement We know \sum_{n=1}^{\infty}\frac{1}{n^x} is uniformly convergent on the interval x\in(1,\infty) and that its sum is called \zeta(x). Proof that \zeta(x) \rightarrow \infty as x \rightarrow 1^+. Homework Equations We cannot find the formula that \zeta is given...
  42. P

    What is infinity to the power of zero?

    Homework Statement Hi everyone, I'm just wondering if someone could please clarify for me what infinity to the power of zero is? I seem to be finding conflicting opinions about this online. Is it '1' or 'not defined'? Thanks! Homework Equations The Attempt at a Solution
  43. R

    Infinity: a concept or number?

    Is infinity a mathematical concept or a number? Please elaborate. Is there any debate over this topic or is there a consensus amongst academia?
  44. J

    Limit as n-> infinity with integral - Please check

    Limit as n-> infinity with integral -- Please check Homework Statement Compute \lim_{n \to \infty} \int_{0}^{1} \frac{e^{x^4}}{n} dx. Homework Equations We can put the limit inside the integral as long as a function is continuous on a bounded interval, such as [0,1]. The Attempt at a...
  45. M

    How can I show that the sup(S)=lim{Xn} and the inf(S)=lim{Yn} as n goes to infinity

    Homework Statement How can I show that the sup(S)=lim{Xn} and the inf(S)=lim{Yn} as n goes to infinity for both of those limits? We are assuming S is a nonempty bounded set that is a subset of the Real numbers. Also, {Xn} and {Yn} are monotone sequences that belong to the the set S...
  46. S

    Help finding the what the limit= as n approaches infinity please.

    Homework Statement The lim as n approaches infinity for 10^n/(n+1)! Homework Equations ? The Attempt at a Solution Normally when there's is a problem with a number n raised to a power over another number n raised to a power I take the ratio of the two to get what the limit...
  47. A

    Difference Between lim(n->inf.) n and Infinity

    Let's consider this problem: 0 × lim(n→∞) n If the limit were evaluated first, it would be 0 × ∞ = undefined. If the limit were rewritten as lim(n→∞) 0n, it would be lim(n→∞) 0 = 0. I cannot tell which is the correct interpretation. It seems that if we consider n to be a finite and...
  48. H

    Cubic polynomial function with 3 real roots; one at infinity?

    Is it possible to have a cubic polynomial (ax^3+bx^2+cx+d) which has three REAL roots, with one of them being +/- infinity? If there is, can you give an example? Thanks!
  49. J

    How Tall is an Infinite Stack of Russian Dolls?

    Homework Statement Your Russian friend gives you a housewarming gift, and you open it up to find a set of Russian Dolls. The first doll is 1 foot tall. Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent. Even when the...
  50. I

    True/false convergence of integral from 1 to infinity

    Suppose h(x) is a continuous function for x > 0. If \int^∞_1{h(x)dx} converges then for constant 0 < a < 1, \int^∞_1{h(\frac{x}{a})dx} also converges. The answer is true. Anyone care to explain why? I would have chosen false, because I was thinking that h(x/a) is larger than h(x) so we...
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