Divergent Definition and 193 Threads

Dear everyone, I have a question on how to show that an integral is divigent. Here is the setup: Suppose that we have the following function ##\sigma(x)=\frac{1}{x^{2-\varepsilon}}## for an arbitrary fixed ##\varepsilon>0.## \begin{equation}...

From my physical problem, I ended up having a sum that looks like the following. S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)} I want to know what is the sum when N \to \infty. Here...

A convergent version ( i.e. convergent in the critical strip) of the traditional series for the Riemann Zeta is derived in the video linked at the bottom. It gives the correct numerical values (at least along the critical line, where I tried it out). But although it works numerically, I'm...

I wonder if the following makes sense. Suppose we want to multiply ##\int_0^\infty e^x dx\cdot\int_0^\infty e^x dx##. The partial sums of these improper integrals are ##\int_0^x e^x dx=e^x-1##. Now we multiply the germs at infinity of these partial sums: ##(e^x-1)(e^x-1)=-2 e^x+e^{2 x}+1##...

Hi, I was recently reading about convergent-divergent nozzles and was wondering about how boundary layers grow in them. Question: How does a boundary layer grow in a convergent duct in subsonic flow? How does this compare to the growth of a boundary layer in a divergent duct in subsonic flow...

I am [working][1] on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\det w=\exp(\operatorname{reg }\ln w)$$ which is analogous to how determinant of a matrix can be...

Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values? I mean, operations on divergent integrals are not a problem, and techniques...

Some popular math videos point out that, for example, the value of -1/12 for the divergent sum 1 + 2 + 3 + 4 ... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on. Is there an elementary way to connect this with the more...

I used a green laser pointer (λ = 532nm). I aimed it at a divergent lens that has a focus of -50cm. The distance from the laser does not play a big role. As a result, a very clear interference pattern with a series of concentric circles appeared on the screen. I did not find a suitable...

Show by contradiction that $$ \sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p} $$ diverges. Which famous result is an immediate corollary?

I have air/gas pressurised gas (pressure is 7 barG). I want to know what are the parameters of the divergent nozzle needed so that the pressurised gas can be released to atmospheric pressure level smoothly and necessary enthalpy conversion can be achieved i.e. the air/gas will accelerate to its...

Hello, guys! I would like to know your opinion and discuss this extension of real numbers: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 In essence, it extends real numbers with entities that correspond to divergent integrals and series. By adding the rules...

Interestingly, If I neglect the ##(-1)^n## or ##(-1)^{n+1}## then apply preliminary test, I could find the limit. Whether the limit is not equal to zero, as in series number 1 and 2, then I can conclude the series is divergent. But, if the limit is equal to zero, as in series number 3, then I...

For the diagram In scalar field theory, I have obtained an integral which looks like $$\int_{0}^{\Lambda} \frac{d^4 q}{(2\pi)^4} \frac{i}{q^2 - m^2 + i\varepsilon} \frac{i}{(p - q)^2 - m^2 + i\varepsilon}$$ I am required to calculate this and obtain the divergent amplitude $$i\mathcal{M} =...

There are meaningful ways to assign values to things like 1 - 1 + 1 + ... or 1 - 2 + 3 - 4 + ... In a similar spirit, is it possible to assign a value to the integral of a function like this: ##f(x)=x*sin(x)## or this one: ##g(x)=Re(x^{1+5i})## (Integrals from some value, say zero, up...

If you calculate the uncertainty of a scalar field in the vacuum state, i.e. ##\langle0\left| \phi^2\right|0\rangle##, you get a divergent integral that comes out to something like $$\frac{1}{4\pi^2}\int_0^\Lambda \frac{k^2 dk}{\sqrt{{m^2}+{k^2}}}$$ Where ##\Lambda## is some momentum cutoff...

I have recently been investigating summing divergent series and zeta function regularization's relation to dimensional re-normalization. Making some progress, but it is a bit slow despite literature being available...

I am trying to compute the cross-section for the diagram below with a divergent triangle loop: where ##X^0## and ##X^-## are some fermions with zero and negative charge respectively. I am interested in low energy limits, so you can consider W-propagator as ##\frac {i\eta_{\mu\nu}} {M_w^2}##...

Hi All Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization...

Homework Statement Prove that (n+1)!/2^n is divergent Homework EquationsThe Attempt at a Solution i know that factorials grow faster than exponentials. But on an exam i would not know how to actually prove that this is divergent other than saying that the numerator is growing much faster than...

I don't understand something, the sum n=1 until infinity of (1/n) is a divergent harmonic series meaning that its sum is infinite right? After reading that i started thinking about the finite volume of the function (1/x) being revolved around the x-axis referred to as "Gabriels horn". They say...

Homework Statement Homework Equations - The Attempt at a Solution Here's my work : However , the correct answer is : Can anyone tell me where's my mistake ?

Homework Statement Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent. Homework Equations We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent. The Attempt at a Solution Applying the ratio test, we find that...

Homework Statement Determine if the improper integral is divergent or convergent . Homework Equations - The Attempt at a Solution When i solved the first term using online calculator , the answer was "The integral is divergent" . However , I got 0 . Where is my mistake ?

Homework Statement ##\sum_{n=1}^{\infty }1+(-1)^{n+1} i^{2n}## Is this series divergent or convergent? Homework Equations 3. The Attempt at a Solution [/B] I tried using the divergent test by taking the limit as ##n## approaches ##{\infty }##, but both ##i^{2n}## and ##(-1)^{n+1}## will...

The hyperboloid with equation: ## z^2=x^2+y^2-1 ##, embedded in standard 3-D Minkowski space ( +, +, - ) so that ## z^2 ## is negative, has positive Gaussian curvature equal to 1 ( as found directly from its metric: ## ds^2 = \sqrt{ -dτ^2+(Coshτ)^2 dθ^2 } ## induced from the ambient Minkowski...

Homework Statement Determine if the series is convergent. Homework Equations ∞ ∑ (((2n^2 + 1)^2)*4^n)/(2(n!)) n=1[/B] The Attempt at a Solution I'n using the Ratio Test and have got as far as (4*(2(n+1)^2+1)^2)/((n+1)((2n^2+1)^2)). I know this series converges but I need to find the...

Homework Statement where the boldface type denotes a vector quantity. In general, the incremental surface dS may be expressed as r dψ R d∅. Examining Figure 1, we see that dS = R2 sin∅ dψ d∅ for this particular nozzle. Substituting Equations 3 and 4 in Equation 2 and integrating ∅ from zero...

Homework Statement Given that ##\{x_n\}## is a bounded, divergent sequence of real numbers, which of the following must be true? (A) ##(x_n)## contains infinitely many convergent subsequences (B) ##(x_n)## contains convergent subsequences with different limits (C) The sequence whose...

It is my understanding that the task of enumerating all of the divergent diagrams in a quantum field theory can be reduced to analyzing a hand full of diagrams (well, at the moment I know that this is at least true for QED and phi^4 theory), and that all other divergent diagrams are divergent...

For a divergent paraxial field like $$E = E_0 e^{-\frac{r^{2}}{w(z)^{2}}} e^{-i(kz - tan^{-1}(\frac{z}{z_{0}}))}$$ What is the direction of the momentum density of the E-field. I have two competing feelings about it. 1) The momentum density should be parallel to the Poynting vector, and since...

Suppose we had an infinite series - z = ∑i = 1 to ∞ ( α1(i)x1 + α2(i)x2 + . . . + αm(i)xm ) - rewritten as the cumulative sequence - z(n) = α1(n)x1 + α2(n)x2 + . . . + αm(n)xm - where the xj are linearly independent and normalized (and serve as a finite basis across the sequence). If all...

The focal of the lens equivalent of two thin lens at distance h is $$1/f=1/f_1+1/f_2+h/(f_1 f_2)$$ Therefore, supposing that ##f_1>0## and ##f_2>0## (both lenses are convergent), if ##f_1+f_2 <h## then the equivalent lens should be divergent. Nevertheless consider the example in picture...

Poster warned that the homework template is not optional. Determine if they are convergent or divergent, If it converges find the sum: ∞ ∑ 3^(n-1) 2^n n=1 ∞ ∑ ln(1/n) n=1 ∞ ∑ tan^n ( π/6) n=1 I tried to find information on how to solve them but I couldn't, thanks for the help

Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...

I have: $$\int_{1}^{2} \frac{1}{x lnx} \,dx$$ I can set $u = lnx$, therefore $du = \frac{1}{x} dx$ and $xdu = dx$. Plug that into the original equation: $$\int_{1}^{2} \frac{x}{x u} \,du$$ Or $$\int_{1}^{2} \frac{1}{ u} \,du$$ Therefore: $ln |u | + C$ and $ln |lnx | + C$ So I need to...

Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...

Homework Statement ∞ Σ (-1)n-1 n/n2 +4 n=1 Homework Equations lim |an+1/an| = L n→∞ bn+1≤bn lim bn = 0 n→∞ The Attempt at a Solution So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo. I found that...

I have this: $$ \sum_{n = 1}^{\infty} \frac{n^n}{3^{1 + 3n}}$$ And I need to determine if it is convergent or divergent. I try the limit comparison test against: $$ \frac{1}{3^{1 + 3n}}$$. So I need to determine $$ \lim_{{n}\to{\infty}} \frac{3^{1 + 3n} \cdot n^n}{3^{1 + 3n}}$$ Or $$...

Question ∞ ∑ tan(1/n) n = 1 Does the infinite series diverge or converge? Equations If limn → ∞ ≠ 0 then the series is divergent Attempt I tried using the limit test with sin(1/n)/cos(1/n) as n approaches infinity which I solved as sin(0)/cos(0) = 0/1 = 0 This does not rule out anything and I...

Why must steady currents be non-divergent in magnetostatics? Based on an article by Kirk T. McDonald (http://www.physics.princeton.edu/~mcdonald/examples/current.pdf), it appears that the answer is that by extrapolating the linear time dependence of the charge density from a constant divergence...

I'm really confused about this test. Suppose we let f(n)=an and f(x) follows all the conditions. When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?

Homework Statement So I've found a ton of examples that show you how to find cauchy principal values of convergent integrals because it is just equal to the value of that integral and you prove that the semi-circle contribution goes to zero. However, I need to find some Cauchy principal values...

pls can u say something on coevolution convergent and divergent evolution and adaptive radiation

Hey guys, could someone help explain to me why the integral of 1/x from -1 to 1 is considered divergent? It would seem as if the area underneath the function cancels out with each other to give you the result of zero, but apparently this is not the case. Thanks!

Hello. I asked my professor and he couldn't figure it out. If train A and B leave the same point at the same time, A traveling 60mph, B traveling 75mph, how long will it take for B to have traveled twice as far as A?

Sorry for the bad English , do not speak the language very well. I posted this to know if the statement or " hypothesis " is correct . thank you very much =D. First Image:https://gyazo.com/7248311481c1273491db7d3608a5c48e Second Image:https://gyazo.com/d8fc52d0c99e0094a6a6fa7d0e5273b6 Third...

Homework Statement I'm trying to find out whether or not this sequence diverges or converges. If it converges, then what's the limit. {4+sin(1/2*pi*n)} The Attempt at a Solution This one is a bit confusing to me since sin oscillates between 1 and -1. So if you plug in (pi*infinity)/2, that...

Consider the two divergent series: $$\sum_{n=k}^{\infty} a_n$$ $$\sum_{n=k}^{\infty} b_n$$ Is it possible for ##\sum_{n=k}^{\infty} (a_n \pm b_n)## to converge?