Convergence Definition and 1000 Threads

Homework Statement Hello. I'm not entirely sure what this question is asking me, so I'll post it and let you know my thoughts, and any input is greatly appreciated. If the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, determine if each of the intervals shown below is a possible...

Homework Statement This is from a complex analysis course: Find radius of convergence of $$\sum_{}^{} (log(n+1) - log (n)) z^n$$ Homework Equations I usually use the root test or with the limit of ##\frac {a_{n+1}}{a_n}## The Attempt at a Solution My first reaction is that this sum looks...

Homework Statement - Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. - Suppose ##(y_n)_n## is...

Hi, I've computed 512 terms of a power series numerically. Below are the first 20 terms. $$ \begin{align*} w(z)&=0.182456 -0.00505418 z+0.323581 z^2-0.708205 z^3-0.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &-1.18398 z^7-0.849919 z^8+2.58123 z^9-0.487307 z^{10}-7.57713 z^{11}+3.91376 z^{12}\\...

On the following page on wikipedia: https://en.wikipedia.org/wiki/Fixed-point_iteration the section "Examples" has a second bullet point, where the author suggests ##q=0.85##, but how did they get this number? I tried googling everything and could not find out how ##q## is determined.

Problem: Let $\left(X, M, \mu\right)$ be a probability space. Suppose $f \in L^\infty\left(\mu\right)$ and $\left| \left| f \right| \right|_\infty > 0$. Prove that $lim_{n \rightarrow \infty} \frac{\int_{X}^{}\left| f \right|^{n+1} \,d\mu}{\int_{X}^{}\left| f \right|^{n} \,d\mu} = \left| \left|...

I tried to derive an equation for one sample mean to converge to another sample mean within a 95% confidence interval, but I know I am wrong. Can someone tell me what I did wrong, and what is the correct formula? Suppose: ##\hat{x_1},\hat{\sigma_1},N## are a sample mean, standard deviation...

Homework Statement Hello, I need some feedback on whether this reasons is correct. consider the series Examine the series for absolute convergence. Homework EquationsThe Attempt at a Solution How I have solved this, using the limit comparison test: we have: introducing we have that...

I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ... I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on pointwise convergence... I need some help to understand the 'mechanics' of Example 6.2.2 (iii) ... Example 6.2.2 reads as...

Ok - anyone that has done basic analysis knows the definition of convergence. The series 1-2+3-4+5... is for example obviously divergent (alternating series test). But wait a minute let's try something tricky and perform a transform on it, (its Borel summation, but that is not really relevant...

For a series to be convergent,it must have a finite sum,i.e.,limiting value of sum.As the sum of n terms approaches a limit,it means that the nth term is getting smaller and tending to 0,but why is not the converse true?Should not the sum approach a finite value if the nth term of the series is...

Homework Statement This is a translation so sorry in advance if there are funky words in here[/B] f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ. Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...

Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...

Homework Statement The series is uniformly convergent on what interval? Homework EquationsThe Attempt at a Solution [/B] Using the quotient test (or radio test), ##|\frac{a_{n+1}}{a_{n}}| \rightarrow |x^2*\sin(\frac{\pi \cdot x}{2})|, n \rightarrow \infty##. However from here I'm stuck...

Homework Statement (FYI It's from an Real Analysis class.) Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent. Homework Equations I know that for an integral to be convergent, it means that : $$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.I can also use the...

Hello! (Wave) I am looking at the following exercise: Let $(a_n)$ be a sequence of real numbers such that $a_{2n} \to a$ and $a_{2n+1} \to a$ for some real number $a$. Show that $a_n \to a$. We are given the sequence $x_n=0$ if $n$ is even, $x_n=1$ if $n$ is odd. Check as for the...

Hello! (Wave) Let $m$ be a natural number. I want to check the sequence $\left( \binom{n}{m} n^{-m}\right)$ as for the convergence and I want to show that there exist constants $C_1>0, C_2>0$ (independent of $n$) and a positive integer $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$...

Hello! (Wave) I want to check as for the convergence the sequence $(a^n b^{n^2})$ for all the possible values that $a,b$ take. I have thought the following: We have that $\lim_{n \to +\infty} a^n=+\infty$ if $a>1$, $\lim_{n \to +\infty} a^n=0$ if $-1<a<1$, right? What happens for $a<-1$ ...

Homework Statement Let ##b_1\in \mathbb{R}## be given and ##n=1,2,\dots## let $$b_{n+1} := \frac{1+b_n^2}{2}.$$ Define the set $$B := \{b_1\in\mathbb{R} \mid \lim_{n\to\infty}b_n \text{ converges}\}$$ Identify the set ##B##. Homework EquationsThe Attempt at a Solution I claim that ##B =...

Homework Statement Determine for which ##r\not = 0## the series ##\displaystyle {\sum_{n=1}^\infty(2+\sin(\frac{n\pi}{3})) r^n}## converges. Homework EquationsThe Attempt at a Solution We have to split this up by cases based on ##r##. 1) Suppose that ##0<|r|<1##. Then...

Homework Statement Show that ##\displaystyle \lim_{n\to \infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n}) = \frac{1}{2}## Homework EquationsThe Attempt at a Solution We see that ##\displaystyle \sqrt{n}(\sqrt{n+1}-\sqrt{n}) - \frac{1}{2} = \frac{\sqrt{n}}{\sqrt{n+1}+\sqrt{n}} - \frac{1}{2} <...

Homework Statement With ##a_1\in\mathbb{N}## given, define ##\displaystyle {\{a_n\}_{n=1}^\infty}\subset\mathbb{R}## by ##\displaystyle {a_{n+1}:=\frac{1+a_n^2}{2}}##, for all ##n\in\mathbb{N}##.Homework EquationsThe Attempt at a Solution We claim that with ##a_1 \in \mathbb{N}##, the sequence...

Homework Statement Find the sum of ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) ## Homework Equations No one. The Attempt at a Solution At first I though it as a telescopic serie: ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) =\ln\left(\dfrac{3}{4}\right) +...

Homework Statement By finding a closed formula for the nth partial sum ##s_n##, show that the series ## s=\sum\limits_{n=1}^{\infty}nx^n## converges to ##\dfrac{x}{(1-x)^2}## when ##|x|<1## and diverges otherwise. Homework Equations Maybe ##s=\sum\limits_{n=0}^{\infty}x^n=\dfrac{1}{1-x}## when...

Homework Statement Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or notHomework EquationsThe Attempt at a Solution looking at ## 1/sin (n) ## by comparison, ##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1## for ##n≥1 ## implying that ##{sin (n)}≤n ##...

Homework Statement Expand ##(1+3x-4x^2)^{0.5}/(1-2x)^2## find its convergence valueHomework EquationsThe Attempt at a Solution on expansion ##(1+3/2x-3.125x^2+4.6875x^3+...)(1+4x+12x^2+32x^3+...)## ##1+5.5x+14.875x^2+42.1875x^3+... ## how do i prove for convergence here?

##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...

Hello everyone! I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...

If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?

What do you think about the claim that \frac{x}{\frac{1}{a} + \frac{x}{b}} \;<\; \frac{2b}{\pi}\arctan\Big(\frac{\pi a}{2b}x\Big),\quad\quad\forall\; a,b,x>0 First I thought that if this is incorrect, then it would be a simple thing to find a numerical point that proves it, and also that if...

Homework Statement interval of convergence for n=1 to inf (x-2)n / n3n Homework EquationsThe Attempt at a Solution i used the ratio test and solved for x and got that the interval of convergence is from -1 to 5. now i have to test the endpoints to determine which ones will make the series...

Homework Statement in title Homework EquationsThe Attempt at a Solution so i know that i have to use the ratio test but i just got completely stuck ((2x)n+1/(n+1)) / ((2x)n) / n ) ((2x)n+1 * n) / ((2x)n) * ( n+1) ) ((2x)n*(n)) / ((2x)1) * (n+1) ) now i take the limit at inf? i am stuck here i...

Hi Physics Forums, I have a problem that I am unable to resolve. The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to...

Homework Statement Homework EquationsThe Attempt at a Solution confused here, so my book seems to be saying that 2*abs(an) converges which i thought was bogus so i went over to symbolab and symbolab is saying it diverges which i agree with. Why is my book saying this? Am i misenterpereting...

Homework Statement ##f(x)=\sum_{n=0}^\infty x^n## ##g(x)=\sum_{n=253}^\infty x^n## The radius of convergence of both is 1. ## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n## 2. The attempt at a solution I got: ## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...

Could someone check if my answers are right and help me with question 5?

Homework Statement Attached I understand the first bound but not the second. I am fine with the rest of the derivation that follows after these bounds, Homework Equations I have this as the triangle inequality with a '+' sign enabling me to bound from above: ##|x+y| \leq |x|+|y| ## (1)...

The problem I am trying determine wether ##f_n## converges pointwise or/and uniformly when ## f(x)=xe^{-x} ## for ##x \geq 0 ##. Relevant equations ##f_n## converges pointwise if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1) ##f_n## converges uniformly if ## \lim_{n...

Given a function in ##f \in L_2(\mathbb{R})-\{0\}## which is non-negative almost everywhere. Then ##w-lim_{n \to \infty} f_n = 0## with ##f_n(x):=f(x-n)##. Why? ##f\in L_2(\mathbb{R})## means ##f## is Lebesgue square integrable, i.e. ##\int_\mathbb{R} |f(x)|^2 \,dx< \infty ##. Weak convergence...

Homework Statement https://imgur.com/DUdOYjE The problem (#58) and its solution are posted above. Homework Equations I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct...

Homework Statement Let ##f## be a real-valued function with ##\operatorname{dom}(f) \subset \mathbb{R}##. Prove ##f## is continuous at ##x_0## if and only if, for every monotonic sequence ##(x_n)## in ##\operatorname{dom}(f)## converging to ##x_0##, we have ##\lim f(x_n) = f(x_0)##. Hint: Don't...

Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...

Homework Statement I have to prove that the improper integral ∫ ln(x)/(1-x) dx on the interval [0,1] is convergent. Homework Equations I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1. The Attempt at a Solution The function can be approximated to ln(x) when it approaches...

Hey! :o Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ? I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or...

I have the following sequence: ##s_1 = 5## and ##\displaystyle s_n = \frac{s_{n-1}^2+5}{2 s_{n-1}}##. To prove that the sequence converges, my textbook proves that the following is true all ##n##: ##\sqrt{5} < s_{n+1} < s_n \le 5##. I know to prove that this recursively defined sequence...

Homework Statement Test the series for convergence or divergence ##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...## Homework Equations rn=abs(an+1/an) The Attempt at a Solution With some effort I was able to figure out the 'n' th tern of the series an = \begin{cases} 2^{-(0.5n+1.5)} & \text{if } n...

Homework Statement Prove rigorously that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##. Homework Equations A sequence ##(s_n)## converges to ##s## if ##\forall \epsilon > 0 \exists N \in \mathbb{N} \forall n \in \mathbb{N} (n> N \implies |s_n - s| < \epsilon)## The Attempt at a Solution Let...

Dear Every one, In my book, Basic Analysis by Jiri Lebel, the exercise states "show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit" My Work: The Proof: Bound The sequence is bounded by 0. $\left|{(n+1)/n}\right|...

I am reading the book "Elementary Real Analysis" (Second Edition, 2008) Volume II by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner ... and am currently focused on Chapter 11, The Euclidean Spaces $$\mathbb{R}^n$$ ... ... I need with the proof of Theorem 11.15 on coordinate-wise...

Homework Statement I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...