Infinity Definition and 970 Threads

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...

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+...

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 [SIZE="5"] ^{lim}_{n\rightarrow∞}\frac{n!}{n^{n}} The Attempt at a Solution I know by just looking at it, n^n Approaches infinity...

Can someone please show me how the formula [SIZE="4"]\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)

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 =...

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

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...

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...

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...

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}} =...

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...

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.

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...

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...

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...

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...

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{∞}{∞} ?

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...

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...

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...

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...

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...

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...

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...

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.

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 -...

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...

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.

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...

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...

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...

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...

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.

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...

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

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...

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

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...

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 +...

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...

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

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}}}\\\\\\...

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...

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...

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

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.

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...

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...

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...

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...