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. 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
  2. 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)
  3. 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...
  4. 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...
  5. 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+...
  6. R

    [Calculus] Sequence Limits: n -> infinity (n/n^n)(Use Sandwich Rule?)

    Homework Statement Use sandwich Rule to find the limit lim n> infinity (a_n) of the sequences, for which the nth term, a_n, is given. Homework Equations ^{lim}_{n\rightarrow∞}\frac{n!}{n^{n}} The Attempt at a Solution I know by just looking at it, n^n Approaches infinity much...
  7. C

    Proving that the sum of (u^x)/x from 0 to infinity = e^u

    Can someone please show me how the formula \sum^{∞}_{0}\frac{u^{x}}{x!} = e^{u} Is derived? Or link me to an explanation. Thanks! http://www.wolframalpha.com/input/?i=sum+of+%28%28a%5Ex%29%2F%28%28x%29%21%29%29+from+x%3D0+to+x+%3D+inf (Just to show you what I'm talking about)
  8. S

    Potential Energy to bring in sphere from infinity

    Homework Statement Given a uniform sphere of mass M and radius R. Use integral calculus and start with a mass dm in the sphere. Calculate the work done to bring in the remainder of the mass from infinity. By this technique show that the self-potential energy of the mass is: P =...
  9. K

    Limit as n approaches infinity

    Homework Statement What is the limit of the given equations as n approaches infinity? (1 + 3n-1)/3n
  10. C

    Integrating a function from 0 to infinity correctly?

    Homework Statement I am trying to integrate the PLanck function to get the Stefan Boltzmann law. After factoring out constants, and substituting x = hv/kT I am left with the following integral: B(T) = ∫ x3/(ex - 1) dx integrated from 0 to ∞ The next step in my notes is that the result...
  11. L

    Sum from 0 to infinity of [3/[(n+1)(5^n)]](x-3)^n HELP

    Sum from 0 to infinity of [3/[(n+1)(5^n)]](x-3)^n HELP :) 1. ∞ Ʃ \frac{3}{(n+1)(5^{n}}*(x-3)n n=0 2. The first question I had to answer was: What is f(3)? I found the first 4 terms to be: 3, 3/10(x-3), 3/75(x-3)2, 3/189(x-3)3 So f(3) equals 3, I'm pretty sure. Because...
  12. N

    Limiting Value of P as t Approaches Infinity

    What is the limiting value of P as t->infinity dp/dt=0.4P(10-P) My attempt at the solution was to serperate the function and get each side in terms of one variable dp/(P(10-P)) = 0.4/dt [-ln(|p-10|/|p|)]/10=0.4t -ln(|p-10|/|p|)=4t take e^ of both sides of equation...
  13. L

    Sum from 1 to infinity of (1+n)/((n)2^n) ~ is this right?

    1. ∞ Ʃ (1+n)/[(n)(2n)] n=1 2. When I see that there is an n as an exponent, I think to do the ratio test. ___________________________________________________________________________ \frac{\frac{1+n+1}{(n+1)(2^{n+1})}}{\frac{1+n}{(n)(2^{n})}} = \frac{n(n+2)}{2(n+1)^{2}} =...
  14. L

    Sum from n=1 to infinity of sqrt(n)/(n^2 + 1)

    1. ∞ Ʃ √(n)/(n2 + 1) n=1 Find if it converges. 2. I'm wondering if I can rewrite this by bringing the n1/2 down to the denominator, making it negative... 1/(n-1/2)(n2 + 1) = 1/(n-1 + n-1/2) = n + √(n) ...And it seems to me that this one would diverge because the n value...
  15. L

    Limit n approaches infinity of (1 + n)^(1/n)

    1. lim (1 + n)^(1/n) n→∞ 2. I was able to figure out that the limit goes to 1 only after I substituted larger and larger values in place of n in my calculator. Since I cannot use a calculator on my test, is there another way to know what the limit goes to? Thanks.
  16. L

    Limit as n approaches infinity of (3n^2 + n + 1)/(5n^3 -2n + 2)

    1. lim (3n2 + n + 1)/(5n3 - 2n + 2) n→∞ 2. In order to solve this problem, do you just think about what happens when n is replaced with a really big number? So, in this case, the numerator only has an n2 and an n, but the denominator has an n3 and an n... So the bottom would always be...
  17. M

    Limits at Infinity: Let S(n) Converge to S

    I have a question about limits at infinity, particularly, about a limit I have seen in the context of infinite series convergence. Let's say we have an infinite series where the the sequence of partial sums is given by {S(n)} and also, it is convergent and the sum is equal to S. Then we know...
  18. J

    Help with optics, telecentric lens, and focusing at infinity

    Hi Guys, I have a question about a telecentric lens I've been studying and was wondering if you guys could help clear some cobwebs in my head. For starters, I am simplifying the TC lens as a compound lens. Also, this lens has 2 degrees of freedom: (1) it can be extended and retracted (2) it...
  19. phoenixthoth

    A statement equivalent to the definition of limits at infinity?

    I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition. So the question is this: are the following two statements equivalent? (1) \lim_{x\rightarrow\infty}f\left(x\right)=L (2) \exists c>0\exists...
  20. U

    Question about PI and infinity

    How comes \frac{\pi}{\pi}= 1 yet \frac{∞}{∞} is indeterminate? I mean \pi is infinite... so it's essentially just another type of infinity. If I said that \frac{3,4,5,6,7...∞}{3,4,5,6,7...∞} = 1 would I be correct? Or again would this be the same as \frac{∞}{∞} ?
  21. E

    Question related to inequalities and limits that go to infinity

    I want to show that if f(x) > g(x) \forall x \in (-\infty, \infty) and \displaystyle\lim_{x\to\infty}g(x)=\infty , then \displaystyle \lim_{x\to\infty}f(x)=\infty . This result is true, correct? If so, what theorem should I use or reference to show this result? I wasn't sure if...
  22. nomadreid

    Chi-squared dist. converges to normal as df goes to infinity, but

    chi-squared dist. converges to normal as df goes to infinity, but... This is surely going to sound naive, but at least this will make it easy to answer. For a chi-squared distribution, if k = the degrees of freedom, then [a] k = μ = (1/2) σ2 [b] as k goes to infinity, the distribution...
  23. ShayanJ

    Is infinity factorial equal to the square root of 2 pi?

    I was studying about infinite products that I got to the relation below in http://mathworld.wolfram.com/InfiniteProduct.html \infty != \sqrt{2 \pi} It really surprised me so I tried to find a proof but couldn't. I tried to take the limit of n! but it was infinity.Also the limit of...
  24. A

    Questions about Attachments: Outer Measure and Infinity

    I have 2 questions about the attachments. 1) In the second attachment, I'm a bit confused about the thing that I marked: O \sim E = \cup^{\infty}_{k=1} O_k \sim E \subseteq \cup^{\infty}_{k=1} [O_k \sim E_k]. I just don't understand how \cup^{\infty}_{k=1} O_k \sim E can be smaller than...
  25. S

    Limit of a certain function of n as n goes to infinity

    Homework Statement \lim_{n->\infty} 3(\frac{n}{n+1})^nThe Attempt at a Solution Ok, I know that the answer is 3/e, because this limit was solved a year ago when I took calculus 1 by my teacher, and I foolishly copied only the answer, thinking I would never forget and have to go back. I can't...
  26. S

    Why doesn't mass of light equal infinity

    I'm trying to wrap my head around what happens as mass is converted to energy. In a nuclear reaction, it is my understanding that mass is converted to energy. It is also my understanding that as matter approaches the speed of light it's mass approaches infinity. If that is so, why is the mass...
  27. M

    Gravity and the Infinity Problem 2 Questions

    I am an avid reader of physics, cosmology, quantum mechanics; the entire genre. I have 2 physics questions: 1) If I understand properly, isn't gravity the effect of a massive object warping the fabric of space-time? If this is correct, then is gravity not really a 'force', but a manifestation...
  28. 0

    Proving the divergent integral of 1/f(x) as x-> infinity

    Homework Statement There exists a function f(x) such that the indefinite integral of 1/f(x) as x-> infinity diverges, and f(x) >= x for all values of x. Prove this function must be a linear polynomial. Homework Equations None that I know of.The Attempt at a Solution No idea where to start.
  29. L

    Infinity potential hole - transition to base state

    Hi! I try to figure out the probability (sure as function of parameter time) of transition particle in the hole from the first excitation state to the base state in an infinity potential hole. Because the eigenfuncion of particle there are orthogonal, the probability looks like zero -...
  30. L

    Finding the limit as x-> infinity: [sqrt(5 + 5x^2)]/(5 + 7x)

    Finding the limit as x--> infinity: [sqrt(5 + 5x^2)]/(5 + 7x) 1. lim[√(5 + 5x^2)]/(5 +7x) x→∞ 2. Alright, I thought I would have first find the largest exponent of x in the denominator. In this case, the largest exponent is x^1. The next step is to divide every term by x^1. Since I...
  31. D

    Can't understand how to compute this limit where x tends to infinity

    Homework Statement This is a question from a past exam paper: Homework Equations The Attempt at a Solution I really had no idea how to approach this but the solution is: Hopefully someone can explain to me the method used to obtain this answer.
  32. R

    How Does an Astronomical Telescope Form an Image at Infinity?

    In astronomical telescopes, they use a convex mirror to from a real image, which is formed at the focus of the eyepiece lens, effectively forming an image at infinity. But how can it truly be at infinity? If it was truly at infinity then how could you see it? Also they say that image at infinity...
  33. O

    Dealing with Infinity: Exploring Theories of Boundless Questions

    How do we deal with the concept on infinity? It seems to linger about the limits of everything. For me, this is the biggest problem in some sense. I wanted to ask the question in the context of the the multiverse. This is a big trend now in physics and cosmology and usually gets lots of...
  34. F

    Focusing at infinity for a camera and circles of confusion

    Hello Forum, when a camera is focused at "infinity", everything from infinity on is in focus (acceptable). How is that possible? Every plane that is not in perfect focus has a certain circle of confusion that gets larger (more blurring) as we move away from the best focus plane... Also...
  35. T

    Gravitational Potential Energy with reference point at infinity?

    So, the gravitational potential energy of a mass "X" from the sun is, let's say, 100joules. Why is it that when we take the gravitational potential energy of the mass from the reference point of infinity that the gravitational potential energy is -100joules? I understand the negative...
  36. V

    Understanding the Infinity Sign for Beginners

    Homework Statement Question is in attachment Can someone explain to me what the infinity sign is I'm new to this topic. All I know is that as a satellite goes to a higher orbit that the velocity decreases.
  37. J

    Exploring Infinity: The Proof and Assumptions

    The proof that 1/2+1/4+1/8+...=1 goes like this: X=1/2+1/4+1/8+... 2X=2/2+2/4+1/8+... 2X=1+1/2+1/4+... 2X-X=1+(1/2-1/2)+(1/4-1/4)+... X=1 The assumption that goes with this is that we can pair up the first term of X with the second term of 2X and so on without having the smallest term of...
  38. S

    Equations of Infinity: Circle to Square

    we have a circle for x^2+y^2=a^2 around the origin. this bulges for x^4+y^4=a^4 this go on for x^n+y^n=a^n as n -> tends to infinity. it actually splits to becomes x=a , x= -a , y= a, y=b which form a square around the origin
  39. ElijahRockers

    Integrating absolute values over infinity

    Homework Statement Find <x> in terms of X0 if X0 is constant and \Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}} and <x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx where Psi* is the complex conjugate of Psi. Since there is no imaginary component, this is effectively Psi2. so, from...
  40. S

    About the properties of infinity

    If we consider a number 9999999………. infinite times then no other number that can be represented bigger than that without plus or multiply operation. So we can be sure infinity is an odd number am I right
  41. B

    Proving the Limit of (3n+5)/2(n+1)^2 is 0 as n Approaches Infinity

    Prove that the limit of (3n+5)/2(n+1)^2 is 0 when n goes to infinity. Attempt: I need to find an N such that for any €>0, (3n+5)/2(n+1)^2<€ holds for every n with n>N. Then I made some manipulation; (3n+5)/2(n+1)^2 < (3n+5)/(2n^2 +4n) < (3n+6)/(2n^2 +4n) = (n+3)/(n^2 +2n) < (n+3)/n^2...
  42. PhizKid

    Solving \lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1}

    Homework Statement \lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1} Homework Equations The Attempt at a Solution \frac{\sqrt{9x^6 - x}}{x^3 + 1} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\ \frac{\frac{\sqrt{9x^6 - x}}{x^3}}{1 + \frac{1}{x^3}} =\\ \frac{(1 +...
  43. W

    Solve a limit with the form (x^n)(e^ax) when x approaches to infinity?

    solve a limit with the form (x^n)(e^ax) when x approaches to infinity? Well, my question is how to solve a limit with the form (x^n)(e^ax) when x approaches to infinity using L´Hopital rule?? I made a try, transforming the limit to (x^n)/(e^-ax), and using L´Hopital repeatedly, gives me...
  44. D

    MHB What is the limit at infinity of (3n+5)/(2n+7)?

    $\lim\limits_{n\to\infty}\frac{3n+5}{2n+7}=\frac{3}{2}$ How does one use a delta epsilon proof for a limit at infinity?
  45. PhizKid

    Limits to Infinity: Solving for $\frac{2x}{\sqrt{x+2} + \sqrt{x}}$

    Homework Statement \lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\ Homework Equations The Attempt at a Solution \lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\ \lim_{x \to \infty} \frac{\frac{2x}{x}}{\sqrt{\frac{x}{x}+\frac{2}{x}} + \sqrt{\frac{x}{x}}}\\\\\\...
  46. qpwimblik

    Why Doesn't Calling 10 Dimensions an Infinity Defeat M-Theory?

    Isn't it so that for 10 dimensions in M Theory to be an actual infinity each of the 3D spatial aspects in each universe have to also be Infinity Large. Thus if each universe is Infinity large would it not contain all possible universes at all possible states in time of All possible universes...
  47. C

    Limit at infinity; l'hopital's rule not working as expected

    Homework Statement Lim(t->(inf)) 1/2((t^2)+1) + (ln|(t^2)+1|)/2 - 1/2 Homework Equations N/A (unless L'Hopital's rule can be counted as an equation for this section) The Attempt at a Solution Background: The problem started with: inf ∫(x^3)/((x^2)+1)^2 dx 0 Using partial fraction...
  48. B

    Limits of sequences as x heads to infinity

    cn= (4n)/(n+4n^(1/n)) When i set it up i think i should use l'hopital but I am confused what to do with the 4n^(1/n) term. an=(7^(2n))/(n!) I know this is a geometric sequence and top and bottom increase initially then tend to 0, but I am lost on how to show the work. should i expand...
  49. T

    MHB Real Analysis is all about infinity

    My lecturer posted a question asking why ""Real Analysis is all about infinity" Why is this so?
  50. L

    Limit approaching negative infinity

    lim x( (x2 −2x+5)^(1/2)−|x−1|) x→−∞ so far, the only way I have started the question is by multiplying for the conjugate but i cannot get it to simply to the answer which is -2 after that step.
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