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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25 views

Uniform convergence of $\sum^n_{k=-n} \frac{1}{z+k}$

Let $D=\mathbb C \setminus \mathbb Z$ and define $$f_n(z)=\sum^n_{k=-n}\frac{1}{z+k}$$ I have to prove that $\{f_n\}^\infty_{n=0}$ is locally convergent on D. We are given the hint to write $f_n$ as a ...
2
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1answer
30 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each ...
3
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1answer
41 views

When does $f_n(x) = a_n \times (1 - nx)$ converge uniformly?

The sequence of functions $\{f_n\}_n$ is defined on $[0,1]$ by: $$f_n(x) = a_n \times (1 - nx),\ {\rm\ if}\ x \in ]0,\frac{1}{n}],$$ and $f_n(x) = 0$ otherwise, where $(a_n)_n$ is a positive ...
1
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1answer
25 views

Can a series of polynomials converge non-uniformly?

Is there an example of a series of polynomials, say, the degree equals the index and converges non-uniformly? In other words, does point-wise convergence of a polynomial series imply uniform ...
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1answer
32 views

Prove that $f_{n}$ is not uniformly convergent

Let $\{f_{n}\}$ be this sequence of functions: $f_{n}(x)=nx$ when $0\leq x\leq \frac{1}{n}$, $f_{n}(x)=2-nx$ when $\frac{1}{n}<x<\frac{2}{n}$ and 0, when $\frac{2}{n} \leq x \leq 1$. I have to ...
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2answers
32 views

If $\{f_n\}$ converges uniformly and each $f_n$ is bounded, show that $\exists M>0$ s.t. $|f_n(x)| \leq M$

Problem Statement: Let $\{f_n\}$ converge uniformly on a set $E$. Suppose that each $f_n$ is bounded. Prove that there exists an $M > 0$ s.t. $|f_n(x)| \leq M$ for all $x \in E$ and all $n = 1,2, ...
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2answers
96 views

Is $f(x) = \sum^{\infty}_{n=1} \sqrt{x} e^{-n^2 x}$ continuous?. Where is bluff?

I have a function defined by $f(x) = \sum^{\infty}_{n=1} \sqrt{x} e^{-n^2 x}$. The task is to check, whether $f(x)$ is continuous at $x = 0$. I have proposition of a solution and I would like someone ...
2
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0answers
42 views

Uniform convergence and equicontinuity

Given a sequence of functions which is not uniformly convergent, can we deduce, that none of its subsequences is uniformly continous and therefore, by Arzela-Ascoli say that the family of function is ...
2
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1answer
35 views

If $\mu$ is $\sigma$ finite and $f_n \rightarrow f$ a.e then $f_n \rightarrow f$ uniformly on each $E_j$

If $\mu$ is $\sigma$ finite and $f_n \rightarrow f$ a.e, there exists $E_1,E_2, \ldots \subset X$ such that $\mu((\bigcup_{1}^{\infty}E_j)^{c})=0$ and $f_n \rightarrow f$ uniformly on each $E_j$ My ...
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1answer
21 views

Uniform convergence with derivative matters

Suppose I have $C^1$ function $f$ such that both $f$ and $f'$ are bounded, and they can both be approximated uniformly by polynomials. My question is, how can one prove that $\exists\,P_n(x)$ which is ...
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1answer
35 views

Uniform convergence of the complex Fresnel integral

Consider the integral $I(\lambda) = \sqrt {\frac {\lambda \mathbb{i}}{\pi}}^n \int_U \mathbb{e}^{-\mathbb{i} \lambda \|x\ - x_0|^2} f(x) \mathbb{d}x, \lambda>0$ and $U$ some open neighbourhood of ...
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1answer
10 views

Uniform convergence of decreasing functions on increasing finite sets

Let $f_k$ be a function from the finite set $S_k$ to the real interval [0,1], with $S_k\subseteq S_{k+1}$. Let also $S=\bigcup_{k\ge 1} S_k$ and assume that $S$ is the set of rational numbers in ...
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2answers
39 views

Prove the sequence converges uniformly and satisfies f(x)=1/(1-x)

Let $[a,b]\subset (-1,1)$, and define the sequence $f_n:[a,b] \to \mathbb{R}$ by $f_n(x)=x^n$. Prove that $\sum_nf_n$ converges uniformly, and that $f= \sum_nf_n$ satisfies $f(x)=1/(1-x)$. Here's ...
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1answer
34 views

Uniformly convergence of series on compact set

Prove that the series summation $$ \sum_{n=-\infty}^\infty \frac{1}{(z-n)^2} $$ converges uniformly on compact sets. I am struggling on this problem in complex analysis. I just know uniformly ...
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0answers
34 views

Local Uniform Convergence

Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as ...
0
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1answer
176 views

A conjecture on uniform convergence of functions with a compact metric space

So I was having a discussion with a friend about this problem and we have conflicting views. Here it is We let $f_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ and we ...
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0answers
60 views

proving point wise convergence but no uniform convergence on f

Let f$_n$: E → $R$ be continuous functions for 1 ≤ n ≤ N. Let a$_k$$^n$ be N convergent sequences of numbers and assume $\lim_{k \to inf}$ a$_k$$^n$ = a$_n$. Let f = $\sum_{n=1}^N$a$_n$f$_n$. I am ...
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1answer
32 views

Prove that $f$ is bounded

Let ${f_n}$ be a sequence of bounded functions $f_n:S\to \mathbb R$ that converge uniformly to $f:S\to \mathbb R$. Prove that $f$ is bounded. Does this prove the statement and are there any flaws in ...
3
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1answer
53 views

$\sum_{n=1}^{\infty}\frac{\sin (nx^2)}{1+n^3}$ represents a differentiable function

Show that the following series of function defines a continuous differentiable function function in $\mathbb R$. $$\sum_{n=1}^{\infty}\frac{\sin (nx^2)}{1+n^3}.$$ We have , $|f_n(x)|=\left|\frac{\sin ...
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1answer
38 views

Show that the series converges point wise and converges uniformly.

I have been attempting quite a few uniform convergence and point wise convergence questions to help prepare myself for an exam coming up in April. I haven't been getting to far on these particular ...
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2answers
35 views

Is the convergence pointwise or uniform?

For the sequence of functions, $\{f_n\}$ $n>-1$ Let $f_n:[0,\infty)\to \mathbb R$, and let it be defined by $f_n(x)=\frac{x}{1+n+x}$. (1) Does $f=\lim_{n\to \infty} f_n$ exist? (2) If this ...
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2answers
33 views

test the uniform convergency of the sequence of function

Test the uniform convergency of the following sequence of functions in $[0,\pi]$. $$ f_n(x)=\frac{\sin nx}{1+nx}$$ Clearly we can see that in converges pointwise to zero funcion in $[0,\pi]$. But ...
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2answers
49 views

If $u_n \rightarrow u$ in $C([a, b])$, is it true that $\int u_n(x) dx \rightarrow \int u(x) dx$?

Let $u_n \rightarrow u$ in $C([a, b])$.Is it true that $$ \underset{a}{\overset{(a+b)/2}{\int}} u_n(x) dx \rightarrow \underset{a}{\overset{(a+b)/2}{\int}} u(x) dx ?$$ If $u_n \rightarrow u$ in ...
2
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1answer
42 views

Question about uniform convergence of a series of functions.

I have encountered a problem which I think is troublesome. Let $g_n,g \in C^0 [a,b],n \ge 1$ such that $g_n,g$ are monotonically increasing functions. Let $X \subset [a,b]$ be an arbitrary dense ...
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0answers
23 views

uniform convergence and improper integral(convolution)

I want to show that $$\frac{d}{dx}(f*g)=(\frac{d}{dx}f)*g$$ where $f(x)=\frac{1}{\sqrt{x}}e^{-\frac{1}{x}}$, $g(y)$ is continuous and bounded. the convolutions are improper integrals. I'm now here ...
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1answer
27 views

Testing for Uniform Convergence of the sum of an Alternating Series.

I'm still trying to get used in understanding the concept behind uniform convergence, so there's another questions which I'm currently have trouble trying to answer. Suppose there's a series ...
3
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2answers
50 views

Testing a series for uniform convergence using Weierstrass' M test

I'm currently having some trouble trying to test for uniform convergence of the series. $\sum_{k=0}^{\infty}\frac{1}{kx+2}-\frac{1}{kx+x+2}$ $0 \leq x \leq 1 $ I tried to test for uniform ...
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3answers
51 views

Test the uniform convergency of a series of function

Consider the series of function $$\sum_{n=1}^{\infty}\frac{x}{1+n^2x}.$$ Show that this series of function is NOT uniformly convergent in $[0,1]$. I know only two methods to show a series of ...
3
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1answer
74 views

The Set of Functions satisfying $\int_{D}\vert f(z)\vert (1-\vert z\vert)^2dA(z)\le 1$ is a Normal Family

Let $\mathcal{F}$ be a family of holomorphic functions on the unit disc so that for any $f\in \mathcal{F}$ one has $$\int_{D}\vert f(z)\vert (1-\vert z\vert)^2dA(z)\le 1$$ Prove $\mathcal{F}$ is ...
1
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1answer
33 views

When converges of function imply converges of derivative?

Let $F_n$ be a sequence of differentiable real valued functions. Suppose that $$\lim_{n \to \infty} F_n(x) = F(x)$$ and that $F(x)$ is differentiable. Under which conditions does that imply ...
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1answer
26 views

Uniform convergence in series definitions of functions

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!
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0answers
30 views

Prove that in any set $A_\epsilon=(-\infty,-\epsilon]\cup[\epsilon, \infty)\subset \mathbb{R}$…

Prove that in any set $A_\epsilon=(-\infty,-\epsilon]\cup[\epsilon, \infty)\subset \mathbb{R}$, where $\epsilon>0$, the convergence of $\forall x \in \mathbb{R}\:\: \lim_{n\to\infty} ...
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3answers
28 views

Interchange of sum and limit in sequence algebra

As you may know, Let $ {a_n} $ and $ {b_n} $ be convergent sequence with limit L, M respectively, then the following is true $ \lim_{n\to \infty} (a_n + b_n) = \lim_{n\to \infty} a_n +\lim_{n\to ...
3
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1answer
67 views

Checking convergence of $\sum\frac{\sin nx}{n}$

Consider the sequence $$f_n(x)=\sum_{k=1}^{n}\frac{\sin kx}{k}\quad x\in \mathbb{R}$$ now we have to check convergence of $\{f_n\}$. Now, I used Dirichlet's criterion to show that $f_n$ converges ...
2
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1answer
47 views

Sum of a geometric series of functions: $\sum_{n=0}^\infty\frac{x^2}{(1+x^2)^n}$

$$\text{Let }f_n(x)=\frac{x^2}{(1+x^2)^n} \text{ for } x\in\Bbb{R}$$ $$\text{I found }\sum_{n=0}^\infty f_n(x) = x^2+1 \text{ since it is a geometric series}$$ Now I'm asked to find $a<b$ such ...
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0answers
25 views

Find an upper bound for $ x$

If $f_1(x)=x, f_2(x)=x^x, f_{n+1}(x)=x^{f_n(x)}$ for $x \geq 1$ and $n \geq 1$. How do you find an upper bound for $x$ as $n \to \infty$ where f_n(x) exists.
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1answer
54 views

Find $f(x)$ from $f_n(x)$.

So I am learning about uniform convergence, pointwise convergence. In order to show uniform convergence, you must show $|f_n(x) - f(x)| = 0$, but I am really confused on how to find f(x) especially ...
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1answer
40 views

Find where this series uniformly converges [closed]

Given the following series: $$f_n(x) = \frac{x^2}{(1+x^2)^n}$$ for $x\in\mathbb{R}$, and let $s_k = \sum_{n=0}^kf_n(x)$. Find values $a < b$ where the series uniformly converges on $[a, b]$. So ...
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2answers
38 views

Uniformly converge with equicontinuous family

Let $\{f_n\}$ be a equicontinuous sequence and converge pointwise in a compact set $K$ of $\mathbb{R}^n$. Prove that the sequence converge uniformly in $K$. My attempt: Since $\{f_n\}$ is ...
0
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1answer
44 views

Prove that ${f_n}$ converges uniformly to the zero function

Fix $a,b∈R$ with $a<b$. Define the sequence ${f_n}$ of functions by $f_n:[a,b]\to\mathbb{R}$ by $f_n(x)=x/n$. Prove that ${f_n}$ converges uniformly to the zero function. I somewhat understand how ...
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votes
4answers
47 views

Prove ${f_n}$ does not converge uniformly to the zero function.

Let ${f_n}$ be a sequence of functions $f_n:(0,1)\to\Bbb R$ defined by $f_n(x)=1/(nx)$. Prove that ${f_n}$ does not converge uniformly to the zero function. If you could walk me through this to ...
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1answer
67 views

Mean value theorem in this uniform convergence proof

I want to prove this theorem: Let $(f_n)$ be a sequence of differentiable functions defined on the closed interval $[a, b]$, and assume $(f_n')$ converges uniformly on[a, b]. If there exists a point ...
3
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0answers
95 views

uniform convergence of a series of sine functions

How would you go about proving that $\sum\limits_{k}\frac{1}{k}\sin\pi\left(\frac{k^2}{x+k}\right)$ converges uniformly in ($0$, $\pi/2$) I'll appreciate any help.
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2answers
70 views

Uniform convergence on $[0,1]$

Let $f_n(x)=x^n$. The sequence $\{f_n(x)\}$ converge pointwise but no uniformly on $[0,1]$. Let $g$ be continuous on $[0,1]$ with $g(1)=0$. Prove that the sequence $\{g(x)x^n\}$ converge uniformly on ...
1
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1answer
65 views

Uniform convergence and differentiable functions proof

If $f_n \to f$ is pointwise on $[a.b]$ and each $f_n$ is differentiable, and $(f'_n)$ converges uniformly on $[a,b]$ to some function $g$, I want to prove that $f'=g$. So if we let $\epsilon > 0 $ ...
2
votes
3answers
43 views

Show that $\frac{1}{n} \sum_{k=1}^n f_k$ converges uniformly to $f$. [duplicate]

Let $f_n:[0,1]\to [0,1]$ continuous functions and let $f:[0,1]\to [0,1]$ such that $f_n$ converges uniformly to $f$. Show that $\frac{1}{n} \sum_{k=1}^n f_k$ also converges uniformly to $f$. Now, ...
0
votes
0answers
27 views

$\sum_{n\ge0}f_n$ converges locally uniformly on $U$

How can I conclude that $\sum_{n\ge0}f_n$ converges locally uniformly on $U$, with $U=\mathbb C\setminus\mathbb R_-$ $f_n(z)=\frac{(-1)^n}{z+n}, \quad z\in U$ I've already proved that ...
0
votes
2answers
35 views

Uniformly convergent sequence of functions in (0,1) that do no converges uniformly in [0,1]

Does there exist a uniformly convergent sequence of functions in (0,1) that does not converges uniformly on [0,1]?
2
votes
1answer
33 views

Show $\Sigma_{k = 0}^\infty e^{-kx}$ converges uniformly on any closed subinterval of $(0,\infty)$

Prove that $\Sigma_{k = 0}^\infty e^{-kx}$ converges uniformly on any closed subinterval of $(0,\infty)$ attempt: by the Weierstrass M-Test if $I$ is a nonempty subset of $ R$ and let $f_k: I → R$ ...
1
vote
1answer
34 views

Series and uniform convergence

Let $\displaystyle \sum_{n=1}^{\infty}a_n $ be a series of real numbers that converges then prove that: the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^x} $ converges uniformly on $[0, ...