Convergence Definition and 1000 Threads

Questions; For the three definitions quoted in the below Background section, I would like to know what the subtle differences are. I know they all have to do with extracting diagonal sequence from a bunch of sequences and its convergence. Especially with Definition 2., why it seems to be...

## Question: For the following Exercise:, (assuming the two exercises in the Assumed Exercises: under the Background below: I thought I am ask to show that if I was able to define a metric on the convergence satisfying properties (1) and (2), that the convergence would violate the following...

I'm reading a proof of a lemma that $$A_rf(x)=\frac1{m(B(r,x))}\int_{B(r,x)}f(y)\,dy,$$where ##m## is Lebesgue measure, is jointly continuous in ##r## and ##x## (##A## stands for average). The claim that ##\chi_{B(r,x)}\to\chi_{B(r_0,x_0)}## on ##\mathbb R^n\setminus S(r_0,x_0)## is made in the...

I will create my own example on this- Phew atleast this concepts are becoming clearer ; your indulgence is welcome. Let me have a sequence given as, ##Un = \dfrac {7n-1}{9n+2}## ##Lim_{n→∞} \left[\dfrac {7n-1}{9n+2}\right] = \dfrac {7}{9} ## Now, ##\left[ \dfrac {7n-1}{9n+2} - \dfrac {7}{9}...

How do I know whether or not the series $$ \sum_{n=1}^\infty sin(\frac{n\pi}{2})sin(nx)$$ converges pointwise for all real x or not? By the way am I right in thinking that converging pointwise for all real x means whatever x i plug into the series it converges to some finite value? I was...

Obviously, you can tell from the fraction that it converges. My problem is their explanation of this process in the book is extremely convoluted, so I'm not too sure what to do with this? From what I gather from their example in the book, I'd want to first create ##b_n## out of the "important...

Theorem 1. If a series ##{a_n}## converges, then the sequence ##{a_n}## converges to ##0##. Now, the contra does not apply, and my question is why? i.e if the the sequence ##{a_n}## converges to ##0## then the series may or may not converge correct? and if it does not converge to ##0## then it...

I am trying to follow this link: https://people.math.wisc.edu/~angenent/521.2017s/UniformConvergence.html ...not getting it... of course i know what convergence is, just to mention for e.g given a sequence, ##\dfrac{1}{n}## I know that the sequence tends to ##0## or rather converges to ##0##...

Here is a cute calculation about which I have my doubts: Treating the derivative as a limit makes the first step a case of switching the order of limits. One cannot automatically do this, as for example for the sequence of functions: More precisely, that one should be able to switch limits iff...

For this problem, Let ##a_n = \frac{1}{n(\ln n)^p}## ##b_n = \frac{1}{(n \ln n)^p} = \frac{1}{(n^*)^p}## We know that ##\sum_{2 \ln 2}^{\infty} \frac{1}{(n^*)^p}## is a p-series with ##n^* = n\ln n##, ##n^* \in \mathbf{R}## Assume p-series stilll has the same property when ##n^* \in...

In the photos are two proof questions requiring proving convergence of sequence from convergent subsequences. Are my proofs for these two questions correct? Note in the first question I have already proved that f_n_k is both monotone and bounded Thanks a lot in advance!

Hi, I'm having difficulty understanding why the interval of convergence is (0, 18]. When I tested x=18, I got the following conclusion using the ratio test. When I attempt using AST, the function still diverges as the lim (n -> inf) = 2^n / n ≠ 0. What am I missing? Thanks!

Hello, this is my attempt for #19 for 11.6 of Stewart's “Multivariable Calculus”. The question is to determine whether the series is absolutely convergent, conditionally convergent, or divergent. The answer solutions used a ratio test to reach the same conclusion but I used the comparison test...

I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... . It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... . Any hints on how to compute them?

For this problem, The solution is, However, does someone please know why this did not use ##2n ≤ 2n^2 + 2n + 1## which would give ##\frac{3n - 1}{2n^2 + 2n + 1} ≤ \frac{3n}{2n} = \frac{3}{2}##? In general, after solving many problems, it seems that when proving the convergence of a rational...

For this problem, I am confused how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here is the first method: $$-1\frac{1}{2} < x - 4 < -\frac{1}{2}$$ $$1\frac{1}{2} > -(x - 4) > \frac{1}{2}$$ $$|1\frac{1}{2}| > |-(x - 4)| >...

This recent video on Numberphile revisits -1 / 12 😱 after a hiatus of nearly 10 years. One point that they make is that there are infinitely many choices of regulating function that converge directly to the correct value (e.g. -1/12) without having to throw away "infinities" or terms of order...

Hi, I am having problems with task d) I now wanted to check the convergence using the quotient test, so ## \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1## I have now proceeded as follows: ##\frac{a_{n+1}}{a_n}=\frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{3^{k+1}} \cdot...

For an example of a Picard iteration, see here. In this case, we have \begin{align} &x_0(t)=x(0)=0,\nonumber\\ &x_1(t)=x_0(t)+\int_0^t \big(1+(x_0(s)-s)\big)^2ds=t+\frac{t^3}{3},\nonumber \\ &x_2(t)=x_0(t)+\int_0^t \big(1+(x_1(s)-s)\big)^2ds=t+\frac{t^7}{3^27},\nonumber\\ &\cdots \nonumber...

I'm writing code to numerically solve a single variable equation, currently with Newton Raphson's method. Right now, I'm just using an initial guess of 1, and reporting a failure if it doesn't converge. While it usually works, it does of course fails for many functions with asymptotes or other...

Let ##\{a_n\}_{n = 1}^\infty## be a sequence of real numbers such that for some real number ##p > 1##, ##\frac{a_n}{a_{n+1}} = 1 + \frac{p}{n} + b_n## where ##\sum b_n## converges absolutely. Show that ##\sum a_n## also converges absolutely.

Let ##\{X_n\}## be a sequence of integrable, real random variables on a probability space ##(\Omega, \mathscr{F}, \mathbb{P})## that converges in probability to an integrable random variable ##X## on ##\Omega##. Suppose ##\mathbb{E}(\sqrt{1 + X_n^2}) \to \mathbb{E}(\sqrt{1 + X^2})## as ##n\to...

Hello! I'm trying to model an interaction between ligands and a heme group using FMO in Gamess. I've tried to make HMOs for the FMOBND section of the input file (using an Fe-F complex), the HMOs are shown below: STO-3G 19 5 0 1 0.992624, 0.019366, 0.000000, 0.000000,-0.000001,-0.014838...

For what values of β the following series converges $$ \sum_{k=1}^{\infty} k^{\beta} \left( \frac{1}{\sqrt k} - \frac{1}{\sqrt {k+1}}\right) $$ I thought of doing it like this $$ \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta} }{\sqrt{k+1}}$$ $$0 \lt \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta}...

##a_n= \left[\dfrac {\ln (n)^2}{n}\right]## We may consider a function of a real variable. This is my approach; ##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]## Applying L'Hopital's rule we shall have; ##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[...

Prove that if ##\{X_n\}_{n = 1}^\infty## is a sequence of real random variables on probability space ##(\Omega, \mathscr{F},\mathbb{P})## such that ##\lim_n \mathbb{E}[X_n] = \mu## and ##\lim_n \operatorname{Var}[X_n] = 0##, then ##X_n## converges to ##\mu## in probability.

Greetings According to my understanding: if x converges in 4 means that the series converges -1<x+3<7 but the solution says C Any hint? thank you!

I'm computing the trajectory of a moving body and my net is composed by 5 stations. My observations are DTOA: difference in time of Arrival (they have been linearized). I am trying to use Least Squares with a linear model: Y = Ax + b, where Y are the observed measurements (DTOA), A the design...

Intertropical Tropical Zone is the zone where north-east and south-east trade winds converge. This zone usually occurs over (I don’t know if “on” should be here) the equator. In the book The Atmosphere: An Introduction to Meteorology by Lutgens and Tarbuck (13th Edition), Figure 7.9 reads I...

Dear Everybody, I have a quick question about the \M\ in this proof: Suppose \b_n\ is in \\mathbb{R}\ such that \lim b_n=3\. Then, there is an \ N\in \mathbb{N}\ such that for all \n\geq\, we have \|b_n-3|<1\. Let M1=4 and note that for n\geq N, we have |b_n|=|b_n-3+3|\leq |b_n-3|+|3|<1+3=M1...

Use a numerical method to solve a PDE f[u(x),u'(x),...]=0, where f is an operator, e.g. u'(x)+u(x)=0, and obtain a numerical solution v(x). Define f[v(x),v'(x),...] as the residual of the original PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution...

Greetings all I have a question regarding the convergence criteria ratio, abs(an+1/an) or the n√abs(an) when the limit tend to a value less than 1 does it mean the serie is convergent or absolutely convergent? Thank you!

Greetings! I have a question about one assumption regarding this question even though I agree with the answer but I have a doubt about A, because when we study the convergence of a serie we use the assymptotic approximation, so why A is not correct? thank you! when we

I would like some clarity on the highlighted part. My question is, consider the the attached example ##(c)##, This sequence converges ( by using L'Hopital's rule)...now my question is, the sequence is indicated on text as not being monotonic...very clear. Does it imply that if a sequence is not...

Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!

I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...

I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is: X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ). Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s). If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s...

If there are points A and B in space, If an object travels a distance from A to any other point between A and B, does that count as converging? And if the object from point A reach point C (a point between A and B) without traveling a distance, does that count as converging too? Like if the...

Hey! :giggle: We have the sequence of functions $$f_n=\sin (x)-\frac{nx}{1+n^2}$$ I want to check the pointwise andthe uniform convergence. We have that $$f^{\star}(x)=\lim_{n\rightarrow \infty}f_n(x)=\lim_{n\rightarrow \infty}\left (\sin (x)-\frac{nx}{1+n^2}\right )=\sin(x)$$ So $f_n(x)$...

Hey! :giggle: For $n\in \mathbb{N}$ let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ given by $f_n(x)=\frac{x+2n}{x^2+n}$. (a) Determine all (local and global) extrema of $f_n$ and the behaviour for $|x|\rightarrow \infty$. Make a sketch for $f_n$ and $f_n'$. Show that there exists $x_1<x_2<x_3<x_4$...

Greetings here is the exercice My solution was as n^2+n+1/(n+1) tends asymptotically to n then the entire stuffs inside the sinus function tends to npi which make it asymptotically equal to sin(npi) which is equal to 0 and consequently the sequence is Absolutely convergent Here is the...

Greetings I have some problems finding the correct result My solution: I puted Y=6x+16 so now will try to find the raduis of convergence of Y so let's calculate the raduis criteria of convergence: We know that Y=6x+16 Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not...

Let ##\epsilon>0##. Then there is an integer ##N>0## with the property that for any integer ##n\geq N##, ##|a_n-A|<\epsilon##, where ##A\in\mathbb{R}##. If for all positive integers ##n##, it is the case that ##|a_n-A|<\epsilon##, then the following must hold: \begin{eqnarray}...

I'm not sure which test is the best to use, so I just start with a divergence test ##\lim_{n \to \infty} \frac {n+3}{\sqrt{5n^2+1}}## The +3 and +1 are negligible ##\lim_{n \to \infty} \frac {n}{\sqrt{5n^2}}## So now I have ##\infty / \infty##. So it's not conclusive. Trying ratio test...

The author start of with $\frac{1}{(y+\sqrt{3})^2} - 2 \cdot \frac{1}{1 + y^2} \left( \frac{y}{\sqrt{1+y^2}} \right) = 0$ and arrives at the equation $y = \frac{(1+y^2)^{3/2}}{2(y+\sqrt{3})^2}$ The solution is merely by iterating (use an initial guess value of y, calculate the RHS, then use this...

Hello everyone, I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem. Assume that we are...

So, I know the direct definition of the Laplace Transform: $$ \mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...

Good day I have a question about the convergence of the following serie I understand that the racine test shows that it an goes to 2/3 which makes it convergent but I also know that for a sequence to be convergent the term an should goes to 0 but the lim(n---->inf) ((2n+100)/(3n+1))^n)=lim...