Convergence Definition and 1000 Threads

Test for convergence: $\sum_1\sin\left(\frac{1}{n}\right)$ My attempt: $\sin\left(\frac{1}{n}\right)\sim\frac{1}{n}$ Since $\sum_1\frac{1}{n}$ diverges, $\sum_1\sin\left(\frac{1}{n}\right)$ also diverges by the asymptotic convergence test. Is that correct?

Let $\{a_n\}$ be a sequence, and $\{a_{n_i}\}$ be any subsequence. Prove that if $\sum_{n=0}^\infty a_n$ is absolutely convergent, then $\sum_{i=0}^\infty a_{n_i}$ is absolutely convergent. My attempt: $\sum |\ a_n|$ is convergent. $b_n=\left\{ \begin{array}{rcl}|a_{n_i}|\ &\text{for}& \...

To talk about differentiable vector fields in Galilean space-time, one needs to define convergence. Galilean space-time is an affine space and its associated vector space is a real 4-dimensional vector space which has a 3-dimensional subspace isomorphic to Euclidean vector space. There is no...

Prove: if $\sum a_n$ is absolutely convergent and $\{b_n\}$ is bounded, then $\sum a_nb_n$ is convergent. My working: $|b_n|\le B$ for some $B\ge 0$. $|a_n||b_n|<B|a_n|$ Since $\sum|a_n|$ converges, $\sum|a_n||b_n|=\sum|a_nb_n|$ converges. So, $\sum a_nb_n$ converges. (Absolute convergence...

Homework Statement If f(z) = \sum an(z-z0)n has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z0| < r ≤ R, show that a0 = a1 = ... = 0. Homework Equations The Attempt at a Solution I know it is a power series and because R is positive I know it converges. And if...

Homework Statement Let (a_n)_{n\in\mathbb{N}} be a real sequence such that a_0\in(0,1) and a_{n+1}=a_n-a_n^2 Does \lim_{n\rightarrow\infty}na_n exist? If yes, calculate it. Homework Equations The Attempt at a Solution I have a solution but I'd like to see other solutions..

Hello, I am struggling to understand why L1 is not weakly compact, while Lp, p>1, is. The example I have seen put forward is the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, the function being defined on (0,1). It is shown this u_n converging to the Dirac measure, and...

Homework Statement If (an) — > 0 and \bn - b\ < an, then show that (bn) — > b Homework Equations The Attempt at a Solution

Homework Statement Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. a) lim \frac{1}{6n^2+1}=0 b) lim \frac{3n+1}{2n+5}=\frac{3}{2} c) lim \frac{2}{\sqrt{n+3}} = 0 The Attempt at a Solution A sequence a_n...

Homework Statement Homework Equations The Attempt at a Solution I have no ideas how to continue. I also tried the comparison test but I don't know where to start. Please guide me...

Hello!A little problem:With the given series,$$Y(x)= \sum_{n=1}^{\infty}(-1)^n\frac{x^n\ln^nx}{n!} $$ ,why $Y(x)$ is Uniformly converges for all $x\in(0,1]$ ?Ok, I know that $Y(x)$ is u.c by M-test: $$\max{|x\ln{x}|}=\frac{1}{e}$$ And, $$ \sum_{n=0}^{\infty}\frac{(\frac{1}{e})^n}{n!} $$ Is...

\sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n z in C. This only converges for z>\frac{-1}{2}, correct? Thanks.

Homework Statement If {S_n} is a sequence whose values lie inside an interval [a,b], prove {S_n/n} is convergent. We don't know Cauchy sequence yet. All we know is the definition of a bounded sequence, and convergence and divergence of sequences. Along with comparison tests and Squeeze...

Homework Statement Determine whether the following sequence {xn} converges in ℂunder the usual norm. x_{n}=n(e^{\frac{2i\pi}{n}}-1) Homework Equations e^{i\pi}=cos(x)+isin(x) ε, \delta Definition of convergence The Attempt at a Solution I would like some verification that this...

\sum_{n=2}^{\infty}z^n\log^2(n), \ \text{where} \ z\in\mathbb{C} \sum_{n=2}^{\infty}z^n\log^2(n) = \sum_{n=0}^{\infty}z^{n+2}\log^2(n+2) By the ratio test, \lim_{n\to\infty}\left|\frac{z^{n+3}\log^2(n+3)}{z^{n+2}\log^2(n+2)}\right| \lim_{n\to\infty}\left|z\left(\frac{\log(n+3)}{ \log...

Hey All, I have the following integral expression: y = lim_{h\to0^{+}} \frac{1}{2\pi} \left\{\int^{\infty}_{-\infty} P(\omega)\left[e^{i\omega h} - 1 \right] \right\} \Bigg/ h And I am trying to understand when this expression will be zero. I was talking to a mathematician who said...

Homework Statement Hey, so here is the problem: Suppose 0≤c<1 and let an = (1 + c)(1 + c2)...(1 + cn) for integer n≥1. Show that this sequence is convergent. Well I understand the basic concepts of proving convergence of sequences, but in class we've only ever done it with sequences where...

Let C(X) be the set of continuous complex-valued bounded functions with domain X. Let {fn} be a sequence of functions on C. We say, that fn converges to f wrt to the metric of C(X) iff fn converges to f uniformly. By definition of the uniform convergence, for any ε>0 there exists integer N...

Homework Statement I have a solution to the following problem. I feel it is somewhat questionable though If fn converges uniformly to f, i.e. fn\rightarrowf as n\rightarrow∞ and gn converges uniformly to g, i.e. gn\rightarrowf as n\rightarrow∞ , Prove that fngn...

Homework Statement \sum\frac{1}{n^{p}} converges for p>1 and diverges for p<1, p\geq0. The Attempt at a Solution (1) Diverges: I want to prove it diverges for 1-p and using the comparison test show it also diverges for p. \sum\frac{1}{n^{1-p}}=\sum\frac{1}{n^{1}n^{-p}}=\sum...

Homework Statement a) Test the following series for convergence using the comparison test : sin(1/n) Explain your conclusion. Homework Equations The Attempt at a Solution i must show f(x)<g(x) in order for it to converge other wise divergence. g(x) = 1/n sin(1/n) >...

So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such. I've come to understand that this property follows...

Not sure where to post about measure theory. None of the forums seems quite right. Suppose that ##(X,\Sigma,\mu)## is a measure space. A sequence ##\langle f_n\rangle_{n=1}^\infty## of almost everywhere real-valued measurable functions on X is said to converge in measure to a measurable...

There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D...

Folks, For C[0,2*pi] and given a function f(x)=sin(x) the supremum |f(x)|=max|f(x)| for x in [a,b] I calculate the sup|f(x)| to be = 0 but my notes say 1. The latter answer would be the case if f(x) was cos(x)...right?

Homework Statement Use a valid convergence test to see if the sum converges. Ʃ(n^(2))/(n^(2)+1) Homework Equations Well, according to p-series, I'd assume this sum diverges, but I don't know which test to use. The Attempt at a Solution I probably can't do this, but I was think...

Hi, I'm using the finite difference method to solve both the Reynolds Equation and the Poisson equation in order to calculate the air pressure of a porous air bearing. The Reynolds equation describes the flow in the channel between the rotor and stator, while the Poisson equation the flow...

Is the sum of 2 divergent series Ʃ(an±bn) divergent? From what I have learned is that it is not always divergent. Is this true? I believe that is what the picture i included is saying, but i maybe miss interpreting it. Also, is the product of 2 divergent series divergent or convergent?

Let (Xn) be a sequence satisfying |Xn+1-Xn| ≤ λ^n r Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.

Hi guys, Problem: Let {Xn},{Yn} - real-valued random variables. {Xn}-->{X} - weakly; {Yn}-->{Y} weakly. Assume that Xn and Yn - independent for all n and that X and Y - are independent. Fact that {Xn+Yn}-->{X+Y} weakly, can be shown using characteristic functions and Levy's theorem...

I am a little confused as to notation for convergence. I included a picture too. If you take a look it says "then the series Ʃan is divergent" Does the "Ʃan" just mean the convergence as to the sum of the series, or the lim an as n→ ∞ nth term? I believe it is the sum of the series but I...

Finding the radius of convergence... Homework Statement 1+2x+(4x^(2)/2!)+(8x^(3)/3!)+(16x^(4)/4!)+(32x^(5)/5!)+... Homework Equations I would use the ratio test. Which is... lim as n→∞ (An+1/An) The Attempt at a Solution I know what to do to find the answer, but I don't know...

Homework Statement Ʃ((x-3)^(n)) / (n*2^(n)) Homework Equations lim as n→ ∞ (An+1 / An) The Attempt at a Solution When dividing two fractions, invert the second and multiple to get what you see below. (x-3)^(n+1)/((n+1)*2^(n+1)) * (n*2^(n))/((x-3)^(n)) Do some cross...

Suppose I have the Laurent series with region of convergence given below: f(z)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad \sqrt{3}<|z|<\sqrt{5} Can I conclude the Laurent-Puiseux series: f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n has a region of convergence...

I am able to use a variety of methods to check to see if a series converges, and I can do it well. However, it's not something I feel like I've intuitively conquered. I don't understand why the series 1/x diverges. I mean, I do, in that I know the integral test will give me the limit as x ->...

Homework Statement Show that given some ε > 0, there exists a natural number M such that for all n ≥ M, (a^n)/n! < ε Homework Equations The Attempt at a Solution Ok so I know this seems similar to a Cauchy sequence problem but its not quite the same. So I am looking for a...

Homework Statement Find the function that (x^n)/(1 + x^n) converges to as n goes to infinity, on the interval [0,2] Homework Equations The Attempt at a Solution I've worked out the fact that on the interval [0,1) it converges to 0, and when x is 1 it converges to 1/2, but for the...

Let A = {(x,y) in R^2 | x^2 + y^2 <= 81} Let B = {((x,y) in R^2 | (x-10)^2 + (y-10)^2 <= 1} then here "A intersection B" is the empty set. Then let x_n be the sequence (0,10-(2/n)) which is a sequence in A and y_n be the sequence (10/n,10) which is a sequence in B. would |x_n - y_n| tend...

Good evening, I'm an electrical engineering student questioning my answer to this series Region of Convergence problem. Ʃ(0,inf) (n(n-1)(z+5i)^n)/n Using the ratio test lim n-> |an+1/an| I was able to get it down to lim n->|n(z+5i)/(n-1)| which gave |inf/inf| = 1, which means the test...

I've seen this thread: https://www.physicsforums.com/showthread.php?t=297842 and that is the exact question I need to to answer. What is the radius of convergence of 1/(1+x^2) expanded about x_0=1? The problem is, I can only use an argument in real analysis. I see the answer is...

Homework Statement This is supposed to be really easy, but I don't think my answer is good Consider this \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} I was hinted that a_{n + 1} = \sqrt{1 + a_n} for all n ≥ 0 and I am supposed to show that the sequence convergees The Attempt at a...

Ʃ ,n=1,∞, (2/n^2+n) Does this series converge or diverge? Im not sure how to start can i use the comparison test here?

Homework Statement Determine if the following series converges or diverges. If it converges determine its sum. Ʃ1/(i2-1) where the upper limit is n and the index i=2 Homework Equations The General Formula for the partial sum was given: Sn=Ʃ1/(i2-1)=3/4-1/(2n)-1/(2(n+1) The...

Homework Statement Determine whether (-1)^n/ln(n!) is divergent, conditionally convergent, or absolutely convergent. Homework Equations None, really? :SThe Attempt at a Solution Okay, so I know the series converges by the Alternating Series test - terms are positive, decreasing, going to...

Homework Statement For x,y \in\mathbb{R} define a metric on \mathbb{R} by d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) | where \tan^{-1} is the principal branch of the inverse tangent, i.e. \tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2). If (x_n)_{n\in\mathbb{N}} is a sequence in \mathbb{R} and...

Homework Statement Find the interval of convergence of each of the following Ʃ^{∞}_{n=0} (\frac{3^{n}-2^{2}}{2^{2n}}(x-1)^{n}) Homework Equations Please refer to attachment The Attempt at a Solution Please refer to attachment. All I want to know is that I'm doing this problem...

I was reading this article of wikipedia: Conditional and absolute convergence It says: "An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long." Is that a characterization of...

Homework Statement Ʃ^{∞}_{n=1} \frac{sin(1/n}{\sqrt{n}} Homework Equations The Attempt at a Solution lim_{n→∞} \frac{sin(1/n}{\sqrt{n}}= lim_{n→∞} \frac{-2cos(1/n)}{n^{3/2}} with l'hospital rule = 0 since lim_{n→∞}=0 therefore the series is convergent Do you think I did this...

Homework Statement Suppose a sequence (f_n)_{n\in\mathbb{N}} converges to a limit f in the metric space (C[a,b],d_{\infty}) (continuous real valued functions on the interval [a,b] with the uniform metric.) Show that f_n also converges pointwise to f; that is for each t\in [a,b] we have...

Homework Statement I am trying to prove the sum of a geometric series, but one of the steps involves deriving this result: \lim_{n\to\infty}r^{n}=0 so that you can simplify the sum of a geometric series, where I have got to this stage: S_{\infty} = \frac{a(1-r^{\infty})}{1-r}...