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.
Homework Statement
Hi there, question asks "What is the difference between the sum to ten terms and the sum to infinity. a = sqroot 2 r = sqroot 2/2
The sum to ten terms, I worked out as 31 + 31 sqroot 2
The sum to infinity, I worked out as 2 sqroot 2 + 2
Homework Equations...
An object at the focal length distance from the lens is imaged at infinity,Do this mean that under this situation, our eyes could not see the image? but as our eyes could see the stars from infinity, do this mean that the image of lens which is discussed above is just viewed through screen ,and...
Hi,I have no idea on how to begin with this question.The question is:
Prove that (n+a)!/(n+b)! ~ na-b as n goes to infinity.There are clue given that we can use Euler's limit and Stirling's formula to solve this question.Can you please give me some hints on how to start with this question...
I'm currently reading a book, The Road To Reality by Roger Penrose, and trying to tackle some of the exercises in the process. My knowledge in mathematics is limited, but broad enough to complete some of the exercises. Anyway, one of them wants you to consider the one function...
lim n-> infinity of:
(n^4 + n^2 + 1)^0.5 - n^2 -1
sin(2/n)/(1/n)
(ln(n) + e^n)/(2^n + n^2)
If anyone could explain the processes required to obtain the limits of any.. (or all) of these statements, that would be great
Try drawing this mentally:
Start with a circle of radius r, draw n number of points spaced evenly on the circle. at each point on the circle draw another circle of radius r, once again with n number of points. What sort of a picture would one get repeating this process a million times, and as n...
I'm 16 and in high school. I've never taken physics or anything before.
I have this equation that shows that pi = infinity times zero
Ok.
So if u take a circle with a diameter of 1 and draw equal spaced radius' and connect them you get a polygon with all equal sides.
the more...
Homework Statement
Hi all.
We we look at z\rightarrow \infty, does this include both z=x for x \rightarrow \infty AND z=iy for y\rightarrow \infty? So, I guess what I am asking is, when z\rightarrow \infty, am I allowed to go to infinity from both the real and imaginary axis? If yes, then this...
Homework Statement
lim_x->∞ of [√x + 2|x|] / [1 + x]
Homework Equations
lim_x->∞ of 1/[x^n] = 0
The Attempt at a Solution
Hi everyone,
Here's what I've done so far:
I divided top and bottom by the highest power of x, i.e. x.
So:
[x^-1/2 +/- 2] / [1/x + 1]
And taking...
Potential difference is what which makes sense. But calculating the potential difference b/w two points and one of them being infinity, where we define potential to be zero, we can actually find the potential at the point. Fine, agreed. Now, how do you define the potential at infinity to be...
Infinity can't be a constant because ∞ ± k = ∞ , but a constant changes (its) value when something is added or subtracted.
But infinity can't be variable because the definition of variable is
"A variable is a symbol that stands for a value that may vary" or stating in simple terms
"In...
Is this true? n,m\in\mathbb{N}
\lim_{n\to\infty}\sqrt[n]{\lim_{m\to\infty}\frac{1}{m}}=\lim_{n\to\infty}\lim_{m\to\infty}\sqrt[n]{\frac{1}{m}}=\lim_{u\to\infty}\sqrt[u]{\frac{1}{u}}=1
Thanks
Homework Statement
X \geq Y > 0, find \lim_{n \to \infty} \left(\frac{2X^n + 7Y^n}{2}\right)^{1/n}
Homework Equations
The Attempt at a Solution
I'm not really sure how to do it, but i guess I need to use the fact that \frac{Y}{X} \leq 1 , and so \lim_{n \to \infty}...
how do i find the limit for the following , where n->infinity
1/2 + 3/4 + 5/8 + 7/16 +9/32 +...
i see that the numerator starts at 1 and has jumps of +2, giving me all the odd numbers
the denominator starts at 2 with jumps of *2 giving all the powers of 2
so i have... + (2n-1)/2^n...
Hi all
This is my first post so please be gentle with me!
Limit of this rational function as x approaches infinity?
f(x) = (x^3 - 2x)/(2x^2 - 10)
I was under the impression that if the degree of the polynomial of the numerator exceed that of the denominator then there could be no...
Homework Statement
Evaluate the integral.
http://www.freeimagehosting.net/uploads/176e17ca2f.jpg
Homework Equations
http://www.freeimagehosting.net/uploads/3168397520.jpg
The Attempt at a Solution
http://www.freeimagehosting.net/uploads/3168397520.jpg
I am stuck here as the book...
why is it that we can use the residue at infinity on the methods of contour integration example on wikipedia for the last one? we can only use the residue at infinity when it is a rational function, i.e. the ratio of two polynomials, if I'm wrong when i say this, why?
I was reading the wiki on this and found it very interesting and would like to hear what established mathematicians and physicists think about this kind of philosophy.
Personally I believe that God(used in reference to the universe itself, am not religious) created infinity and man created...
Let me first say I am in high school and don't understand very much calculus or advanced physics. But through my independent work i have created a theory in which i personally can not find any flaws with. I would ask for constructive critisms on this theory. thanks!
It is my understanding...
I need to find the limit x -> infinity of the following:
y = x ( (2x/a) / (1 + (2x/a)) )
Simplifying..
y = x ( (2x/a) / ((a + 2x)/a) )
y = x ( 2x / (a + 2x) )
y = 2x^2 / (a + 2x)
Is this even right in the first place? because I have no idea how to evaluate the lim x -> infinity.
limit as X--> infinity of cos(x)
Homework Statement
Find the limit.
\stackrel{lim}{x\rightarrow\infty}cos(x)
The Attempt at a Solution
As best as I can tell by putting larger and larger numbers in my calculator, there is no limit, as cos(x) just oscillates between (-1,1). So is the...
Homework Statement
Find the limit as x approaches infinity of (sqrt(1+3x^2))/x
The Attempt at a Solution
I tried using l'hopital's rule, but it gave me 3x/(sqrt(1+3x^2)) which doesn't help me at all.
so i understand how to resolve a limit at x->oo, but from a conceptual standpoint, i do not get it. for example,
limit x->oo, 4x/5x
so the answer is 4/5, but oo/oo is an indeterminate expression
i understand that if i treat x as a variable, then it makes sense, but still
if the example was...
Homework Statement
integral [from e to infinity of ] 67 / (x(ln(x))^3)
read as the integral from e to infinity of
67 divided by x times cubed lnx.
Homework Equations
The Attempt at a Solution
i know it converges,
but i got the value
67/2
Homework Statement
\sum(\frac{1}{\sqrt{ln k +2}-\sqrt{ln k -2})}k
as k \rightarrow\infty
Homework Equations
Root test: (ak)1/k
The Attempt at a Solution
(ak)1/k = (\frac{1}{\sqrt{ln k +2}-\sqrt{ln k -2})
does it equal 0? since 1/\infty = 0
but its \infty - \infty i would have to...
1. My problem is such:
Find the limit of \lim_{x \rightarrow \infty} \sqrt{9x^2+x} -3x
2. No relevant equations
3. I multiplied \frac{\sqrt{9x^2+x} -3x}{1} * \frac{\sqrt{9x^2+x} +3x}{\sqrt{9x^2+x} +3x} = \frac{x}{\sqrt{9x^2+x} +3x}
I am now quite confused as to where to go...
Homework Statement
\lim_{x \to -\infty} x + \sqrt{x^2 + 6x}
Homework Equations
The Attempt at a Solution
Previous attempt was guessing it was \infty, but I see now my flaw and the actual answer is -3. Somewhere else on the web, might have been this forum, it was said that one could flip the...
Homework Statement
I have a space M which is a sequeces of real numbers \{x_n\} where
\sum_{n = 1}^{\infty} x_{n}^2 < \infty
How can a series mentioned above be become than less than infinity??
Please explain :confused:
Sincerely
Cauchy
Suppose we have the following shape in the complex plane: The EXTERIOR of the semi-infinite strip bounded by 0 < y < 1 and x > 0. The two physical angles making up the rectangle have interior angles of 3*pi/2 and thus exterior angles of -pi/2.
Now, because the sum of the exterior angles of a...
Homework Statement
Prove that lim n \rightarrow\infty 2^{}n/n! = 0
Homework Equations
This implies that 2^{}n/n! is a null sequence and so therefore this must hold:
(\forall E >0)(\existsN E N^{}+)(\foralln E N^{}+)[(n > N) \Rightarrow (|a_{}n| < E)
The Attempt at a Solution...
Homework Statement
What is the limit as x goes to positive infinity of:
(4x2+3x+8)/(6x2+5x-7)?
Homework Equations
The Attempt at a Solution
Usually I would factor the top and bottom and see if something cancels, but that doesn't work here.
Also l'hopital rule doesn't work...
Homework Statement
For the following function decide whether f(x) tends to a limit as x tends to infinity. If the limit exists find it.
Homework Equations
f(x)=[xsinx]/[x^2 +1]
The Attempt at a Solution
I thought about using L'Hopitals rule, so i got:
[sinx + xcosx]/[2x]...
In Wheeler and Taylor's 'Exploring Black Holes', on pages 3-12 and 3-13, the bookkeeper measure of radial velocity (i.e. radial velocity as measured from infinity) is derived. Basically the equation for 'Energy in Schwarzschild geometry' is established-...
"If X is non-negative, then E(X) = Integral(0 to infinity) of (1-F(x))dx, where F(x) is the cumulative distribution function of X."
============================
First of all, does X have to be a continuous random variable here? Or will the above result hold for both continuous and...
Homework Statement
Prove that the following sequence (a(n)) has the property that a(n) tends to infinity as n tends to infinity.
Homework Equations
a(n)=[n+7]/[2+sin(n)]
The Attempt at a Solution
i tried l'hopitals rule, so i got 1/cos(n)...which wouldn't work.
so I am not...
I have a system transfer function
H(s) = 1/(e^s + 10)
This system has both poles and zeros at infinity and -infinity.
Can anybody tell me if this is a stable system. Thanks.
Homework Statement
let f(x)= (x^2)/(1+x) for all x in [ifinity, 0) proof that f(x) is uniformly continuous. can anyone help me with this problem
Homework Equations
using the definition of a uniform continuous function
The Attempt at a Solution
i did long division to simplify the...
Kronig-Penney potential as spacing --> infinity
Homework Statement
Show that in the limit that the atomic sites of the Kronig-Penney potential become far removed from each other (b-->infinity), energies of the more strongly bound electrons (E<<V) become the eigenenergies k1a=n*Pi of a 1D...