What is Infinity: Definition and 983 Discussions

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. adjacent

    How can we create an image at infinity using a converging lens?

    https://dl.dropboxusercontent.com/u/260388836/index.html Let arrow height be 1cm. focal length:1cm So if I move the arrow to the focal point,image is not formed.(i.e formed at infinity) But, Lets move it to say 0.999 Now the image is virtual and is magnified to about 1000X If I make it...
  2. L

    How can ln be used to solve a sum to infinity problem?

    Homework Statement Prove the following result: \frac{1}{2.2} + \frac{ 1}{3.2^2} + \frac{1}{4.2^3} ... = 2ln2 -1 Homework Equations The Attempt at a SolutionI tried writing down the nth term of the series which is 1/(n+1)2^n But don't know where to move after this.
  3. L

    What is the solution to the infinite series involving factorials and pi?

    Homework Statement Sum to infinity \frac{1}{2!} - \frac{ \pi ^2}{4^2.4!} + \frac{\pi^4}{4^4.6!} ... Homework Equations The Attempt at a Solution I thought the series was similar to the Maclaurin expansion of cos x so I tried putting in x= ∏/4 But I end up with the...
  4. E

    How do you tell if a limit is going to be infinity?

    How do you tell if a limit is infinity, and you should use that approach, or if you should try to factor/multiply by congegate, etc.? Do you use the latter if its 0/0 and the first if its a number/0 when you try pluging in th limit?
  5. FlexGunship

    Astounding: 1+2+3+4+5+ = [not infinity]

    \displaystyle\sum_{n=1}^{\infty}{n} = {-}\dfrac{1}{12} It seems that, regardless of intelligence, this proof (or sum, or proof of sum, or demonstration of sum) rattles some more than others. It is so astoundingly counter-intuitive that embracing it means really leaving your mathematical...
  6. L

    Sum to infinity question (G.P.)

    Homework Statement ΔABC has AB= 8 in, BC= 10in , CA= 6in . AD is the perpendicular from A to BC. DE the perpendicular from D to AB, EF the perpendicular from E to BD and so on. Show that CA + AD + DE+... is a geometric series and find it's sum to infinity. Homework Equations The...
  7. applestrudle

    Integral (e^-ax)sin(bx) from 0 to infinity

    Homework Statement integral (e^-ax)sin(bx) from 0 to infinity Homework Equations The Attempt at a Solution I want to check if my answer is right I got \frac { \frac { -1 }{ a } { e }^{ -ax }\quad sinbx\quad -\frac { b }{ { a }^{ 2 } } { e }^{ -ax }\quad cosbx }{ 1+\frac {...
  8. D

    Fourier Series from m=1 to infinity

    Simple question; Why isn't it \sum am (from m=1 to infinity) Thanks in advance.
  9. L

    Evaluate the limit as x goes to infinity using L'Hospital's rule

    (1) Evaluate the limit as x goes to infinity using L'Hospital's rule: 8xe^(1/x)-8x (2) L'Hopital's Rule (3) How can I use L'Hopital's Rule for this problem if the denominator is 1? Wouldn't that just give me an undefined limit? This may be a pretty stupid question, but I'm new...
  10. V

    Integrating a physical quantity to infinity

    This is something that has bothered me for some time, and I can't seem to find any threads on here about it. In a lot of my undergraduate courses in physics, we talk about integrating something physical to infinity. For example, in electrostatics, we talk about the work needed to assemble a...
  11. O

    Limit as 'constant' approaches infinity

    Homework Statement Determine limA→∞(sinc(x/2A)) Homework Equations I would use L'Hôpital's rule, but I'm not sure if it is valid in this case as the function is that of x, not A. I want to know if it's valid to treat x as constant and take derivatives with respect to A and evaluate as...
  12. T

    Behavior of e^(i*x) at Infinity and Negative Infinity

    Homework Statement delta dirac function(x) * e^(-i*x) at ∞ and -∞ Homework Equations delta dirac(x)*e^(i*x)The Attempt at a Solution I'm wondering how e^(i*x) looks like at infinity/-infinity. I know it has some sort of oscillating property i.e e^(pi*i)=e^(3pi*i). The problem is I'm trying to...
  13. V

    Is x < ∞ always finite? Exploring the concept of infinity and finiteness

    If something is strictly less than something else, is it definitely finite? Because if x < y, and the biggest thing y can be is ∞, then x < ∞, but does that mean x is finite? Couldn't x = (∞ - 1)? Isn't that still technically infinity? What if x = (∞ - ε)?
  14. B

    Limit proof problem as x goes to infinity

    Homework Statement prove that ##\lim_{x \to \infty} \frac{\sqrt{x+1}}{x} = 0## where ##x>0## Homework Equations Definition: Let ##A\subseteq\mathbb{R}## and let ##f:A\rightarrow \mathbb{R}##. Suppose that ##(a,\infty)\subseteq A## for some ##a\in\mathbb{R}##. We say that...
  15. Q

    Infinity over infinity squared

    Homework Statement http://i.minus.com/jbwNvxhmIxeKxB.png The Attempt at a Solution I'm unclear on whether the entire log is being squared or if just the (1-x) term in the denominator is being squared. I know if it's the latter, then the exponent can be moved to in front of the log.
  16. T

    MHB Another two limits at infinity

    I know I am already boring with limits,but i again have two of them to deal with and i don't know how... 1. \lim _{n \to \infty} \frac{5^{n^+1}-2*5^n+5^{n-1}}{3^{n+1}-3^n} 2. \lim _{n \to \infty} \frac{\sqrt[3]{n^4}+\sqrt{n}+1}{\sqrt[6]{n^4}+\sqrt[3]{n}+2}
  17. T

    MHB How do I solve limits at infinity with similar problems on my exam?

    Again i have two similar problems: \lim _{n \to \infty} \frac{3n-\sqrt{4n^2+n}}{3n+\sqrt{4n^2-n}} \lim _{n \to \infty} \frac{\sqrt{9n^2-n}-2n}{\sqrt{9n^2-n}+2n} Those kind of problems will be on my exam,which is very close,and i don't get it how to deal with this limits... I just say...
  18. V

    Infinity -just maths or any physical existence?

    Iam wondering whether 'infinity' has real physical existence or just a mathematical paradox? If it does have a physical existence why don't we come across any quantity which is physically eternal? Someone please help..
  19. P

    Probability & Infinity: All Combinations Equal?

    If there is an actual infinity of throws of two dice, would all combinations have the same probability (that is each combination would happen an infinite amount of times)?
  20. M

    What is 0 multiplied by infinity in limits?

    If you have f(x) = 1/x and g(a) = (cosx - 1)/x and then y = [limx→0 f(x)][limx→0g(x)], the two individual limits equal 0 and infinity, respectively. Since these are limits and only approach these values, would the multiplication of the two limits equal 0, infinity or something else?
  21. S

    Laurent series around infinity

    Laurent series at infinity point I already calculated it, but my work was too long, I really wish to find a shorter route. Calculate the Laurent series of \frac{1}{(z^{2}+1)^{2}} around z_{0} = 0 First, I used simple fractions and I got: \frac{1}{(z^{2}+1)^{2}}=\frac{-i}{4} \frac{1}{z-i} +...
  22. O

    Sum of 1/(n^2) as n goes to infinity

    Homework Statement Prove Ʃ1/(n^2) as n goes to infinity = (∏^2)/8 Homework Equations The Attempt at a Solution No idea how to start. Pls guide. Thanks
  23. M

    Natural log limits as n approaches infinity

    My question is: Show the limit of x_{n}=\frac{ln(1+\sqrt{n}+\sqrt[3]{n})}{ln(1+\sqrt[3]{n}+\sqrt[4]{n})} as n approaches infinity Solution: {x_n} = \frac{{\ln (1 + {n^{\frac{1}{2}}} + {n^{\frac{1}{3}}})}}{{\ln (1 + {n^{\frac{1}{3}}} + {n^{\frac{1}{4}}})}} = \frac{{\ln \left(...
  24. P

    What is the limit at infinity for the given expression?

    1. Evaluate lim x\rightarrow-\infty \sqrt{x^2+x+1}+x.The answer is -\frac{1}{2}. Homework Equations None. The Attempt at a Solution I multiplied by the conjugate first, so it turns into lim x\rightarrow-\infty \frac{(x^2+x+1)-x^2}{\sqrt{x^2+x+1}-x} = lim x\rightarrow-\infty...
  25. I

    Series expansion of an integral at infinity

    Hello, I'm fiddling with Wolfram Alpha and I can't find a definition of what do they mean by the "Series expansion of the integral at x -> inf". In particular, I have two divergent integrals and I am wondering whether their ratio is some finite number. Here it is: \left[\int_0^{\infty}...
  26. S

    Question: Infinity A bigger than Infinity B ?

    Hello everybody, Is it possible that there can be an infinity A that is bigger than Infinity B ? How does this work/not work because i read that infinity is without limit(on wikipedia) so does saying one thing is bigger than another mean that a limit must be set to reach this conclusion or is...
  27. L

    What does vanishing at infinity mean for a topological space?

    If X is a locally compact Haussorff space, then the set of continuous functions of compact support form a normal vector space C_c(X) with the supremum norm, and the completion of this space is the space C_0(X) of functions vanishing at infinity, i.e. the space of functions f such that f can be...
  28. A

    Limit of one to the power of infinity

    Homework Statement Find limit x->0 (f(x))g(x). Here limit x->0 f(x) = 1 and limit x->0 g(x) = ∞Homework Equations I know that limit x->0 (1 + x)1/x = e ... (1) The Attempt at a Solution I have the answer which is elimit x->0 (f(x) - 1) * g(x) But I don't know how to proceed. I have to...
  29. D

    Can infinity be defined rigorously?

    Can "infinity" be defined rigorously? In math, many of the most fundamental theorems and structures rely on the concept of an infinity. For example, we define the irrational (and transcendental) numbers as being the limits of some Cauchy sequence of rational numbers. This limiting process...
  30. E

    Why the curve r(t) approaches a circle as t approaches infinity

    Both statements 1 and 2 are given as an explanation of why the original statement is true, but I don't understand why you can use statement 2 (since in the original vector equation you do not have Sin2(t), -Cos2(t)) Show why r(t) = <e-t, Sin(t), -Cos(t)> approaches a circle as t →∞. 1. As...
  31. twoski

    What are the Constants c and C for Infinity and One Norms Inequality?

    Homework Statement Determine constants c and C that do not depend on vector x but may depend on the dimension n of x, such that c ||x||_{∞} ≤ ||x||_{1} ≤ C||x||_{∞} Use this result and the definition of matrix norms to find k and K that don't depend on the entries of matrix A (but...
  32. N

    Fortran Solving Negative Infinity in Fortran Calculation

    Dear all, I can not figure out why I got negative infinity in my output, so please help. Here is my calculation for FAMAX: FAMAG=SQRT(FX**2+FY**2+FZ**2) FAMAX=MAXVAL(FAMAG(1:NATOM)) and for FPBPMAX...
  33. M

    How to compute limits at infinity?

    Homework Statement lim x→∞ ##\frac{7x^2 + x + 11}{4 - x}##Homework Equations The Attempt at a Solution I am sorry I am posting so much. But I think I have learned two different ways to compute limits at infinity of functions: one by the math lab tutor and another by the professor, but I am...
  34. A

    Can Relativity and Quantum Theory be Applied to Infinity?

    To those who have seen Toy Story please forgive title. This subject has I believe turned at least one mathematician mad, so I am hoping there are some maths people out there willing to take a chance. We need a starting point to get things moving either a line, a 2d surface or a 3d space as...
  35. C

    What is the Result of 0/0? Exploring Infinity

    If both numbers approach 0 but one does not know their exact state, doesn't that mean the result can be either 1 or anything around it up to -∞and +∞? Does that in turn mean that 0/0 = Everything?
  36. S

    Fortran [Fortran90] fdtd in polar coordinates, got infinity output

    hi all, attached here is my code for 2d fdtd in polar coordinates, from 'numerical electromagnetic: the fdtd method (umran s inan, pg 94-96) written in fortran90. I have try a few approach I could think about to troubleshoot this code but the output is still infinity. Anybody here can give me...
  37. G

    Do quantum fields vanish at infinity?

    In Srednicki's textbook (chpt 5) he has an expression: \int d^3k f(k) \int d^4x (\partial^2 e^{ikx}) \phi(x) and he wants to integrate by parts in order put the Laplacian on the field \int d^3k f(k) \int d^4x (e^{ikx})\partial^2 \phi(x) instead of the the exponential. He says that...
  38. E

    How can a particle coming from infinity get on bound orbit around BH?

    Studying the movement of a particle on bound orbits around a black hole I found a fact that seemed a little strange for me -- that on these orbits particle's total energy on infinity must be less than unit (E<1). As far as I understrand, it is not so horrible, since here we deal with an...
  39. A

    Why Does Tangent Go To Infinity?

    Why does tangent go to infinity when it increases 0 to 90 degrees?
  40. J

    Why limit n-> infinity (3/4)^n = 0

    Why limit n--> infinity (3/4)^n = 0 Homework Statement Quick question. Brain flagrance. Homework Equations Why limit n--> infinity (3/4)^n = 0 The Attempt at a Solution ?? Why is this. How do you know.
  41. R

    Integration of mixed function at infinity limit

    Hi, I am trying to calculate second virial coefficient from interaction potentias and I have to Integrate at Infinity level and it seems that Integrate doesn't converge. can you help me integrate this function. Integrate is attached as an image. Regards Raymond
  42. N

    Problems with limits at infinity within improper integrals

    Homework Statement ##\int_{2}^{\infty} ue^{-u} du## The Attempt at a Solution What I did was find the family of functions described by the indefinite integral ##\int ue^{-u} du## then found the limit as b increases without bound. $$=\lim_{b\rightarrow \infty}...
  43. A

    Calculating Work to Push a Charge from Infinity

    I have just read a thread here:https://www.physicsforums.com/showthread.php?t=382031 Here is part from the thread. Please see the bold sentence in the quote. In this case, does it mean that my force(push) is larger than the push force from fixed charge. If so, how can I calculate the work that...
  44. A

    Taylor expansion at infinity of x/1+e^(1/x)

    I have some problems finding Taylor's expansion at infinity of f(x) = \frac{x}{1+e^{\frac{1}{x}}} I tried to find Taylor's expansion at 0 of : g(u) = \frac{1}{u} \cdot \frac{1}{1+e^u} \hspace{10 mm} \mbox{ where } \hspace{10 mm} u = 1/x in order to then use the known expansion of...
  45. T

    Limits at Infinity for the Argument Function in Complex Variables

    ℂI am working on an assignment and have come across a question that I'm not quite sure how to approach. Here it is, with my "solution" and reasoning: "[F]ind the limit at ∞ of the given function, or explain why it does not exist. 24. h(z) = Arg z , z \neq 0" (Complex Variables Second...
  46. Petrus

    MHB How to Evaluate a Limit Involving Infinity using L'Hopital's Rule

    Hello MHB, \lim_{x->\pm\infty}xe^{\frac{2}{x}}-x I start to divide by x and we know that \lim_{x->\pm\infty} \frac{2}{x}=0 with other words we get 1-1=0 but that is wrong, how do I do this :confused: Regards, |\pi\rangle
  47. W

    Maximum angle which a field line starting from Q will end at infinity

    Homework Statement http://i.imgur.com/haX3OW8.png Homework Equations The Attempt at a Solution I guess a) is 1/3 (probably wrong, because I assumed all the field lines that end at -q originates at +3q which is not necessarily true) but I can't figure out b)
  48. Petrus

    MHB Horizontal Asymptote of Inverse Tangent Function

    Hello MHB, I got one question, I am currently working with an old exam and I am suposed to draw it with vertican/horizontal lines (and those that are oblique). f(x)=\frac{x}{2}+\tan^{-1}(\frac{1}{x}) for the horizontel line \lim_{x->\infty^{\pm}}\frac{x}{2}+\tan^{-1}(\frac{x}{2}) Is it enough...
  49. G

    You can't do operations on infinity

    I'm trying to get my words right. They say you can't do operations on infinity. Sorry I don't have an exact quote. But on the other hand you can do calculations involving infinite series. What is the proper way to describe what math can't do with infinity? I want to say something along...
  50. L

    How does this second integral equal +infinity instead of -infinity?

    1. ...How does this 2nd integral diverge to +∞? It seems to me that it would diverge to -∞... :/ lim (\frac{-1}{a - 1} - \frac{-1}{0 - 1}) + lim (\frac{-1}{2 - 1} - \frac{-1}{b - 1}) + lim (\frac{-1}{c - 1} - \frac{-1}{2 - 1}) a→1- b→1+...
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