# Tagged Questions

For questions about uniform convergence of a sequence or a series of functions on a set. Also used with the tag [convergence].

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### Does $\sum \frac{\sin^2(nx)}{n^2x}$ converges uniformly on $(0,\delta]$?

I'm already aware, by Parsevsal's theorem, that $$\sum_{n=1}^{\infty} \frac{\sin^2(nx)}{n^2x}=\frac{\pi-x }{2}$$ is a pointwise convergence on $(0,\delta]$ for all $\delta\in(0,\pi)$. Now I'm ...
Let $(S,d)$ be a complete separable metric space and consider the set $L^1(S,\mathbb{R})$ of functions $f:S \rightarrow \mathbb{R}$ which are 1-Lipschitz, i.e. $\forall x,y \in S: |f(x) - f(y)| \leq ... 0answers 4 views ### Pointwise bounded, non uniformly convergent fails to be equicontinuous Let f_n : I --> R be x^n. The collection F = {f_n} is pointwise bounded but the sequence (f_n) has no uniformly convergent subsequence; at what points does F fail to be equicontinuous? The way I ... 0answers 26 views ### Uniform convergence of${f_n(x)=\int_{0}^{x}g_n(t)}dt$where$g_n(x)=\sin^2(x+\frac{1}{n})$. [duplicate] I need to prove that $${f_n(x)=\int_{0}^{x}g_n(t)}dt$$ where$g_n(x)=\sin^2(x+\frac{1}{n})$is uniformly convergent on$[0,\infty)[0,1]$How can I do this? Is it correct to argue that if ... 1answer 34 views ### Series Uniform Convergence Question Let$M>0$. $$\sum_{n=M}^{\infty} \frac {1}{(x-n)^2}$$ How do I show this converges uniformly for$x ≤ \frac{|M|}{2}$My actual question is how do I determine the interval of uniform convergence ... 2answers 30 views ### Find the set of x where$\sum_{n=0}^\infty \frac{x}{(2nx-x+1)(2nx+x+1)}$Find the set of x where: $$\sum_{n=0}^\infty \frac{x}{(2nx-x+1)(2nx+x+1)}$$ Converges, calculate the sum. And determine where does the series converge uniformly. Would appreciate any help, 1answer 85 views ### Can I avoid the Abel partial summation technique and instead prove uniform convergence in this way? Edit: I am attempting to use summation by parts...and will revise the below proof, as the R.H.S. says nothing more than what the L.H.S. already gives. The problem statement is: For 2 sequences of ... 2answers 81 views ### Cauchy Criterion - Prove$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{x^k}{\sqrt k}$converges uniformly on$[0,1]$$$\sum_{k=1}^{\infty} (-1)^{k-1}\frac{x^{k}}{\sqrt k}$$ I have shown this converges uniformly in$[0,1]$: But my working also implies$\sum_{k=1}^{\infty}\frac{x^{k}}{\sqrt k}$converges uniformly ... 0answers 16 views ### Non-Power Series Examples of Uniform Convergence I have been working with Ross' Elementary Analysis and almost all examples of uniform convergence are either sequences of functions which converge to the zero function or are power series. I believe ... 2answers 33 views ### On a series of functions exercise:$\sum_{n=1}^{\infty} n \ln (1+ \frac{|\sin x|^n}{1+x^n})$. I want to study the uniform convergence of this series of functions: $$\sum_{n=1}^{\infty} n \ln (1+ \frac{|\sin x|^n}{1+x^n})$$ After trying the root test on the sequence formed by moving the$n$... 1answer 31 views ### Prove that for any$x>1$,$\lim_{n \to \infty} \int_0^n t^{x-1}(1-\frac{t}{n})^n dt = \Gamma(x)$Prove that for any$x>1$, $$\lim_{n \to \infty} \int_0^n t^{x-1}(1-\frac{t}{n})^n dt = \Gamma(x)$$ The hint is to use the dominated convergence theorem, so I did let$f(x)=t^{x-1}e^{-t}$and ... 0answers 30 views ### Uniform convergence from pointwise convergence for uniform Lipshitz functions I would like to prove or give a counterexample for the following statement: Let$(S,d)$be a complete and seperable space. We define: $$\mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid ... 3answers 31 views ### Completeness of B(X,R) I'm trying to show B(X,R) the set of all bounded functions from a metric space X to R(space of real numbers) is a complete metric space with respect to supremum metric, if X is complete. So I took a ... 2answers 38 views ### Uniformly convergent and Weierstrass M-test Heres my question: I used The Weierstrass M-test showing that |x^n(1-x)|\le |x|^n(1-x) and$$\sum_{n=0}^\infty |x|^n(1-x) = (1-x)\sum_{n=0}^\infty |x|^n=(1-x)\frac{1}{1-x}=1 < \infty$$So the ... 1answer 33 views ### Let \{a_n\} be decreasing and \geq 0. Let 0<\varepsilon<1/2. Prove \sum a_n\sin(2\pi nx) is unif. converg. in [\epsilon, 1-\epsilon]. I'm doing the following exercise: Let \{a_n\} be a decreasing and \geq 0 sequence. Let 0<\epsilon<1/2. Prove that \sum a_n \sin(2\pi nx) is uniformly convergent in [\epsilon, ... 1answer 29 views ### About the uniform convergence of a function sequence. I'm reading my notes about uniform convergence, and this question came to my mind: Let g:\mathbb{R} \to (0,+\infty) be a continuous function, for example g(x)=\displaystyle\frac{1}{e^{x}}. Let ... 0answers 38 views +50 ### Uniform convergence of Empirical Moment Generating Function In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of n variables X_1,X_2, \dots, X_n as:$$ ... 1answer 35 views ### Pointwise convergence in$C(K)$where$K$is compact hausdorff Let$ \{f_n\}$be a sequence in$C(K)$where$K$is a compact Hausdorff space with$|f_n| \le 1$for$n=1,2,......$. If$f \in C(K)$and$ f_n \to f$pointwise on$K$then show that there exist some ... 1answer 37 views ### Does Pointwise convergence imply$\int_a^b (f_n(t) - f(t))^2 \, \mathrm d t \to 0$If$f_n$and$f$are integrable and$f_n$tends to$f$uniformly on$[a,b]$then$\displaystyle \int_a^b (f_n(t) - f(t))\,\mathrm d t \to 0 $, since $$\int_a^b |f_n(t) - f(t)| \, \mathrm d t \le ... 0answers 34 views ### If f_n converge uniformly in [0,1] to f then f_n^2 converge uniformly in [0,1] to f^2. Prove that if f_n converge uniformly in [0,1] to f, f_n^2 converge uniformly in [0,1] to f^2. f are continuous in the interval [0,1]. 0answers 27 views ### Uniform convergence of a particular series I'd like to determine the convergence of a series:$$\sum _{n=1}^{\infty}\frac{n^4}{n!}(x^{3n}+x^{-3n})$$Whether or not this series converges uniformly, or whether it converges at all. Thanks. 2answers 42 views ### Check that f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1} is continuous or not. Define f:[0,1]\to \Bbb R by$$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}$$Check that f is continuous or not. Attempt:$$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}\\ ... 1answer 22 views ### Power series differentiability at endpoints I have the following problem: Find domain$I$of the function defined by$f(x)=\sum\limits_{n=1}^{\infty}(3^{\frac{1}{n^2}}-1)x^n.$Investigate differentiability of$f(x)$in the interior of ... 1answer 36 views ### Uniform convergence of$\sum_{n=1}^{\infty}x^{n}/n^{2}$One problem saying: Is$\sum_{n=1}^{\infty}1/(x^{2}+n^{2})$uniform convergence? So I solved it using Weierstrass M test since$\mathbb R $is complete and$g_k\leq 1/k^2$and the series consisting ... 0answers 134 views ### Dominated Convergence Theorem Proof for Improper Integrals My attempt at the proof: As fn --> f uniformly on [a,b] where b is arbitrary chosen in (a,infinity) then lim n --> infinity (fn) = f. As [fn(x)] < g(x) for x in [a,infinity) then applying ... 2answers 53 views ### Is$C^1([a,b])$with sup norm a complete metric space? [duplicate] Give$C^1([a,b])$the sup metric induced from$(C^0([a,b]),||.||)$. I want to know if$C^1([a,b])$is complete. My thinking is to show that$C^1([a,b])$is a closed subset of$C^0([a,b])$. Let ... 2answers 22 views ### Sup norm and uniform convergence in$C^0([a,b])$My book says convergence in sup norm$||f_nm-f||\to 0$is equivalent to uniform convergence and this follows immediately from definitions, but I just want to check:$\Rightarrow$If ... 2answers 81 views ### How to fix a proof of Dini's Lemma I am aware of 2 proofs of Dini. One is by contradiction and Bolzano-Weierstrass, and one is by an open covers definition. I decided to try to make a 'direct' proof without using 'every open cover has ... 2answers 25 views ### Sequence of functions converges to a uniformly continuous function This is a problem from Abbott's Analysis: Let$f$be uniformly continuous on all of$\mathbb{R}$, and define a sequence of functions$f_n(x)=f(x+\frac{1}{n})$. Show that$f_n\rightarrow f$... 1answer 19 views ### e/3 argument? Uniform convergence from a topoloical space to a metric space I am trying to prove that if we take a sequence (say fn) in C(X,Y) that converges uniformly to to a function f: X to Y, then f must be an element of the space C(X,Y). Where f moves from a topology to ... 1answer 29 views ### Term by Term Differentiability in the context of Uniform Convergence I'm not sure how differentiability works with uniform convergence. My book says that we can show this (calculation wise) $$\varepsilon (x,a) = \sum_{k=1}^{\infty} E_{k}(x,a)$$ for some$x$and$a$. ... 0answers 24 views ### Series of Functions - Pointwise and Uniform Convergence I'm learning about series of functions and need some help with this problem : Given the series of functions $$\sum_{n=1}^\infty \frac{x}{x^2+n^2}, \; x \in (0, \infty)$$ show that it converges ... 1answer 33 views ### Show that$x-\frac{x^{2}\sin x}{(1+nx^{2})(1+(n-1)x^{2})}$can be simplified to$\frac{|\sin x|}{1+nx^{2}}$I was doing a problem on uniform convergence. The book wishes to prove $$\frac{x^{2}\sin x}{(1+nx^{2})(1+(n-1)x^{2})}$$ uniformly converges to$x$. The first inequality is derived from the book's use ... 1answer 56 views ### In a proof of uniform convergence, how to find$\epsilon$that works for all interval$[a,b]$? I'm not really getting uniform convergence in terms of actual applications, so I would appreciate some help. The way my teacher taught uniform convergence, the only example he used was to show a ... 0answers 22 views ### Derivative of cumulative generating function at zero equals expectation value Let$X$be a random variable with values in$\mathbb{N_0}$. Then we can define the cumulative generating function of$X$via $$F_{X}: (-\infty, 0] \rightarrow \mathbb{R} \quad \quad t \mapsto ... 0answers 22 views ### Show that E=\bigcup_{k=1}^{\infty}E_k, where for each k, E_k is measurable Let {f_n} be a sequence of measurable functions on E that converges to the real valued f pointwise on E. Show that E=\bigcup_{k=1}^{\infty}E_k, where for each k, E_k is measurable, and {f_n} ... 1answer 21 views ### Computation of uniform limits I have the following sequences of functions: f_n(x)=n^3x^n(1-x) on [0,1/2] (conjecture: this converges uniform to 0) f_n(x)=\frac{x}{nx+1} on (0,1) (conjecture: also uniform limit to 0) ... 2answers 20 views ### When does the limit of derivatives coincide with the derivative of the limit function? Thinking about the (probably) well-known fallacy about approaching a unit square diagonal with staircase functions and thus concluding the diagonal length be 2 instead of \sqrt 2 led me to an ... 2answers 47 views ### Show that if f:\mathbb{R}\rightarrow\mathbb{R} is uniformly continuous, then sequence f(x+1),f(x+\frac{1}{2}),f(x+\frac{1}{3}),\ldots Show that if f:\mathbb{R}\rightarrow\mathbb{R} is uniformly continuous, then sequence f(x+1),f(x+\frac{1}{2}),f(x+\frac{1}{3}),\ldots is uniformly convergent. Let g_n(x)=f(x+\frac{1}{n}). A ... 0answers 43 views ### Prove that \sum_{n=1}^\infty(-1)^n\frac{x^2+n}{n^2} converges uniformly, but not absolutely My Work: i) Uniform Convergence (By Weierstrass M-Test): I am attempting to show that the series converges uniformly on the interval I=[-a,a], in ... 0answers 24 views ### The derivatives of Bernstein polynomials for a C^1 function uniformly converge Let f:[0,1]\rightarrow \mathbb{R} be continuous, and suppose that f' is also continuous. Show that$$\frac{d}{dx}B_{n}(.;f)\rightarrow f'$$uniformly. Here B_{n}(.;f) represents the nth ... 2answers 41 views ### Finding a dominating function I have been asked to find a dominating function for the following sequence of functions:$$f_{n}(x)=\frac{x n^{3/2}}{1+n^{2}x^{2}}, x\in[0,1]$$in order to use the dominated convergence theorem. I ... 1answer 41 views ### Showing equality of sequence of holomorphic functions to limit function if converges uniformly locally I do not know how to proof following question: Let D be a domain in C. {{f}_{n}} be a sequence of holomorphic functions on D that converges uniformly locally to a function non-constant f. ... 0answers 31 views ### for which interval I is the series \sum x^n/(1+x^{2n}) uniform convergent? The series sum n from 1 to infinity. I used ratio test to find that only when \lvert x\rvert >1, the series would converge. But I do not know how to approach the uniform convergent part. ... 1answer 47 views ### Let {f_n} be a sequence of continuous real valued functions on [0, \infty), then which of the following is/are true? Let \{f_n\} be a sequence of continuous real valued functions on [0, \infty). Suppose f_n(x)\to f(x) ~~~\forall x\in [0,\infty) and f is integrable. Then \int_{0}^{\infty} f_n(x)dx \to ... 0answers 42 views ### Uniform convergence of simple functions to a bounded function f Let f be a bounded measurable function on E. Show that there are sequences of simple functions on E, \{\varphi_{n}\} and \{\psi_{n}\} such that \{\varphi_{n}\} is increasing, ... 0answers 14 views ### Uniform convergence of 2-norm of a multinomial vector Let (X_1,X_2,\ldots,X_k) be distributed according to a multinomial distribution with parameters (n;p_1,p_2,\ldots, p_k), i.e.$$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} ... 1answer 29 views ### Show that$f_k'(z)=z^{k-1}$does not converge uniformly for$|z|<1$. 1) Show that$f_k(z)=z^k/k$converges uniformly for$|z|<1$2) Show that$f_k'(z)=z^{k-1}$does not converge uniformly for$|z|<1$. My Try: I did part 1. In part 2, I can prove that ... 1answer 24 views ### continuous series Let$(f_n)$be a sequence of continuous functions on$(0, \infty)$satisfying$|f_n(x)| \leq 1$for all$x > 0$and all$n \geq 1$. Show that the function$f(x) = \sum_{n=1}^{\infty} ...
$f_n(x) = \frac{n^2x}{1+n^3x^2}$ I claim the above function is continuous. For $[a, ∞) , a>0$ I have tried and found out that it is pointwise convergent. For uniform convergence, I found out ...