Divergent Definition and 193 Threads

I am to find whether the sum of (n!)/(n^n) converges or diverges. I tried both the limit comparison test, and a regular comparison test. (These are the only types of tests I am allowed to use.) So I tried several approaches: Approach #1: (n!)/(n^n) > 1/(n^n) Normally we use a setup like...

an = [cos (3n) )] /n.cos(1/n) My solution is , I wrote an as liman→∞ [cos (3n) )] /[ cos(1/n) / (1/n) ) . We get liman→∞ cos(3n) / ( 1/0 ) . I think solution of this limit is zero but ı'm not sure cos(3n) .I think cos(3n) as a number and number/infinity is zero .As a result of...

Homework Statement use the comparison theorem to show that the integral of e^(-x^2) from 0 to infinity is convergent.Homework Equations None The Attempt at a Solution In class we have never dealt with using the comparison theorem with the exponential function so I was not sure what I function...

From mathhelpforum.com... A constructive divergent series Let $\displaystyle \{x_{n}\}$ be a positive sequence satisfying $\lim_{\ n \rightarrow \infty} x_{n}= \infty$ Prove that the series $\displaystyle \sum_{n=1}^{\infty} (1-\frac{x_{n}}{x_{n+1}})$ is divergent. Till now no answers have...

Homework Statement Show that the sum of a convrgent sequence and a divergent sequence must be a divergent sequence. What can you say about the sum of two divergent sequences? Homework Equations A theorem in the book states: Let {a_n} converge to a and {b_n} converge to b, then the...

greetings . we have the integral : \lim_{T\to \infty }\int_{2-iT}^{2+iT}\frac{(s-1)^{n}}{s}ds which diverges for every value of n except n=0 if we perform the change of variables : s\rightarrow \frac{1}{s} then : \lim_{T\to \infty...

Homework Statement Suppose that ∑a_n and ∑b_n are series with positive terms and ∑b_n is divergent. Prove that if: lim a_n/b_n = infinity n--->infinity then ∑a_n is also divergent.Homework Equations The Attempt at a Solution Well in attempting to write a viable solution, I have deducted...

Homework Statement I attached the problem to this post. The Attempt at a Solution I was wondering if I could use the limit comparison test for this integral. My professor taught us this test that can be used for series but could it work for improper integrals as well? So what I would...

Homework Statement I'm supposed to answer true or false on whether or not the sequence ((-1)^n * n) tends toward both ±∞ Homework Equations Uniqueness of Limits The Attempt at a Solution I did prove it another way, but I would think that uniqueness of limits (as a definition...

Homework Statement I have to find whether the following is Convergent or Divergent ∑ from n = 1 to infinity 2 / n(2n + 2)^(1/4) Actually it's the fourth root, this is just easier to write. Homework Equations According to the front of the sheet it's a quiz on P-Series and Integral Test I'm...

Homework Statement If Ʃa_n is divergent, the absolute value of Ʃa_n is divergent. True or false. This is the main question I am trying to answer. I should be able to answer this problem on my own, but i ran into a problem that confused me. What I Did So I decided to start this...

Homework Statement Suppose that ∑_(n=1)^∞▒a_n where a_n≠0 is known to be a convergent series. Prove that ∑_(n=1)^∞▒1/a_n is a divergent series. Homework Equations The Attempt at a Solution if ∑_(n=1)^∞▒a_n is convergent then lim n→∞ a_n = 0. thus for some N a_n>1 and a_n...

Homework Statement Me and my friend are debating on wether the follow seris is convergent or divergent. The seris is the sum of (-1)^n-1 * ln(n)/n. Homework Equations p test and comparision tests. And alternating series test The Attempt at a Solution My approach to this problem...

I need to find out if this function is convergent or divergent when finding the limit to infiniti. nsin(npi) How do I solve this? Do I use the squeeze theorem or lhospital rule?

Homework Statement Prove if sequence a_{n} converges and sequence b_{n} diverges, then the sequence a_{n}+b_{n} also diverges. Homework Equations The Attempt at a Solution My professor recommended a proof by contradiction. That is, suppose a_{n}+b_{n} does converge. Then, for...

Hi all I am looking for a simple way to show that the mean of the Cauchy distribution us undefined. This is because this integral diverges: \underset{-\infty}{\overset{\infty}{\int}}\frac{x}{x^{2}+a^{2}}dx Now, I know one proof which replaces the limits of integration with -x1 and x2. After...

what happen if converging rays are made to pass to a convex lens? what happen if divergent rays are made to pass to a concave lens?

(1) Using the Archimedean definition of divergence, prove that if \sum_{i=1}^{\infty }x_{i} diverges to infinity, then so does either \sum_{i=1}^{\infty }x_{2i} or \sum_{i=1}^{\infty }x_{2i+1}. (2) Show an example where \sum_{i=1}^{\infty }x_{i} diverges to infinity but \sum_{i=1}^{\infty...

Homework Statement I know that if a_n := n^{1/n} - 1, then \Sigma a_n is divergent. I know this (by the integral test) because the integral of 2^{1/n} - 1 from 1 to infinity is infinite. However, I want to avoid using non-elementary functions (here, the exponential integral) in my proof that...

Homework Statement Prove that \sum\limits_{n = 0}^\infty {\frac{{\left( { \sqrt{n+1} \right) - \sqrt{n} }}{{\left( {\sqrt{n}} \right)!}}} is divergent Homework Equations The Attempt at a Solution This is an intro to analysis course. We haven't gone over the integral test which would be...

Homework Statement Basically i have got the infinite sum of (n+2)1/2-n1/2 and i think it is divergent ( I hope) but i have no idea how to show it, the ratio test is not helpful and i cannot find anything to compare it with. Thanks in advance.

Homework Statement For what integer k, k > 1, will both sigma n=1 to infinity ((-1)^(kn))/n and sigma n=1 to infinity (k/4)^n converge? A) 6 B) 5 C)4 D)3 E)2 Homework Equations The Attempt at a Solution I tried to use the ratio test and after some simplifying I got (-1)^k (n/n+1)...

Homework Statement Determine if the series is convergent or divergent. \sum x^2e^{-x^2} Homework Equations The Attempt at a Solution x^2e^{-x^2}=\frac{x^2}{e^{x^2}} \lim_{x\to\infty } \frac{(x+1)^2}{e^{(x+1)^2}}\frac{e^{x^2}}{x^2} and since (x+1)^2=x^2+2n+1 and...

i understood the air flow properties variation in convergent divergent / condi nozzle or laval nozzle for subsonic flow based on the formula, for a incompressible flow : density * Area * velocity = constant as the total pressure in the flow is constant dynamic pressure is...

Homework Statement I have been straining to find convergence or divergence of a few infinite series. I have tried everything I can think of; ratio test, root test, trying to find a good series for comparison, etc. Here are the formulas for the terms: #1 1 ------------- (ln(n))^ln(n)...

From the model used in the zeta regularization procedure to give a meaning to divergent series in the form 1+2+3+4+... , we propose a similar method to give a finite meaning to divergent integrals in the form \int_{0}^{\infty}dx x^{m} for positive 'm' in terms of the negative values of the...

Homework Statement I had to solve the integral...After all my work i got -infinity 3 integral sign 1/ [(t-3)^4/3] 1 Homework Equations The Attempt at a Solution -infinity...is this correct? and would this be divergent?

Hello, I have to determine whether the series converges or diverges. It is \Sigma (-1)^n * cos(Pi/n) where n=1 and goes to infinity. First I took the absolute value of the function and got the limit from n to infinity of cos(pi/n) and as a result I got 1 because cos(0)=1. However my...

Is the series convergent or divergent? n=1 summation and it goes to infinity n!/2n!+1 [Infinite series] Homework Equations None. The Attempt at a Solution I have no idea.

Homework Statement As the title says. Homework Equations mentioned in solution The Attempt at a Solution Let Sn = {(1+1)/(1+2) , (1+2)/(1+4), (1+3)/(1+6), ...}. If ∑(1+n)/(1+2n) is convergent, then lim n-->∞ Sn = 0; to put it another way, there exists an N so that whenever n ≤...

i've got the following problem let be the integral \int_{R^{3}} dxdydz \frac{R(x,y,z)}{Q(x,y,z)} here R(x,y,z) and Q(x,y,z) are Polynomials on several variable. Let us suppose this integral is divergent, in order to regularize it i have thought about substracting several terms so...

A sequence \{x_n\} in a metric space (X,d) converges iff (\exists x\in X)(\forall \epsilon > 0)(\exists N \in \mathbb N)(\forall n > N)(d(x_n,x) < \epsilon). Am I correct when I assert that the negation of this is: A sequence \{x_n\} does not converge in (X,d) iff (\forall x\in X)(\exists...

When the weibull shape parameter beta is <1, the pdf is divergent at t=0 due to negative exponent of beta -1. With such a divergent distribution is it meaningful to use Weibull for beta <1?

Why are no DIVERGENT quantities (infinities) in String Theory ?? why String theory is FREE of infinities ?? ... why there are no divergent integrals in string theory whereas in normal Quantum Field theory there are infinities ??

Homework Statement Prove that the \sum1/n is divergent. Does anyone know a simple proof for this. I understand that it does not converge intuitively but I'm not sure how to prove it in symbols. Thank you for your help. M Homework Equations The Attempt at a Solution

Homework Statement int (e^-x)/(x)dx from 0 to infinity Determine if integral is convergent or divergent2. The attempt at a solution I assume because the bottom limit is 0 and there is an x in the bottom of the integral that this is going to be divergent but I still have to use the...

I am struggling to find two divergent series, \Sigmaan and \Sigmabn, such that the series of minimum terms, \Sigmamin{an,bn}, actually converges. A further stipulation is that both an and bn must be positive, decreasing sequences. (Otherwise the problem is trivial, as one could simply...

Homework Statement I am being asked to consider a Dirac spinor with two complex components and the following Lagrangian: L = L_{Dirac}-\stackrel{g}{4}{(\psi\bar{\psi})^{2}} I am asked to derive the Feynman rules for this theory which I can do using the standard methods. However, I am...

Homework Statement Determine if the following is improper and convergent, improper and divergent, or proper \int \frac{dx}{\sqrt[3]{x^2 - 7}} from 8 to infinityThe Attempt at a Solution Since I don't know how to integrate...

I want to calculate the partial wave amplitudes for various processes but get divergent results in certain cases which I assume is related to the propagator going on shell - the forward scattering amplitude is infinite. I guess I have to introduce some sort of cut but don't know how to do this...

I had a question regarding convergent and divergent integrals. I want to know the "exact" definition of an improper integral that converges. Wikipedia states that For a while, I took that as a valid answer and claimed that any integral that has a finite answer must be convergent. However, I...

Homework Statement Determine whether the following series is convergent or divergent: \frac{1}{2^2}+\frac{2^2}{3^3}+\frac{3^3}{4^4}+... I rewrite it as: \sum_{n=1}^{\infty} \frac{n^n}{(n+1)^{n+1}} Homework Equations The Attempt at a Solution I stopped. I can not do anything.

Homework Statement Determine the convergence or divergence of the series. If the series is convergent, find its sum. Justify each answer. (n=1, to infinity) \sum(7/9 + n^5) Help please? I missed a lot of school recently from being sick and need help with this!

Homework Statement Prove that if $\displaystyle\sum_{n=1}^\infty a_n$ converges and $\displaystyle\sum_{n=1}^\infty b_n$ diverges, then $\displaystyle\sum_{n=1}^\infty (a_n+b_n)$ diverges. Homework Equations I know that the limit of {a_n} = 0 because it is convergent, but I can't say...

I'm confused with the following integral. Let a > 1. \int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left[ \log \left| \dfrac{t}{t-1} \right| \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right| This should be the correct result. But I could also...

Homework Statement Suppose that {a_n} is a divergent sequence of real numbers and c \in R, c <> 0. Prove that {c*a_n} diverges. Homework Equations The Attempt at a Solution I have attempted to solve the problem as a proof by contradiction, but am afraid I am leaving something out...

is this trick valid at least in the 'regularization' sense ?? for example \int_{-\infty}^{\infty} \frac{dx}{x^{2}-a^{2}} then we replace thi integral above by \int_{-\infty}^{\infty} \frac{dx}{x^{2}+ie-a^{2}} for 'e' tending to 0 using Cauchy residue theorem i get...

I can't remember much from my intro. analysis class anymore. If you have an infinite series that ultimately converges, can the first few terms diverge (i.e., can they move away from the convergence point)? And if so, how many of these terms can do so? I'm trying to understand how to "get...

Homework Statement Show \sum_{k=1}^{\infty} c\k^(1\div2k, c is an element of [tex]\Re, c > 0, is divergent. Homework Equations 1/n is divergentThe Attempt at a Solution Finding a similar series and doing comparison test, is it right?

How did Euler came up with a solution of 1/4 for the series 1-2+3-4... There is a wikipedia page explaining it, http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7 but I'm not quite following the manipulation of the numbers. How is a 1 pulled out and...