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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2
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0answers
28 views

Square of uniformly convergent functions is not always uniformly convergent

For each $n ∈ N$ and $x ∈ R$ let $f_n(x) = x + \frac{1}{n}$, and for each $x ∈ R$ let $f(x) = x$. Working directly from the definition of uniform convergence (i.e. without using the Uniform Norm ...
3
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1answer
22 views

Showing that sequences of functions converge uniformly

Let $A$ be a set, let $f,g : A → R$, and let $(f_n)_{n∈N}$ and $(g_n)_{n∈N}$ be sequences of functions from $A$ to $R$. Show that if $f_n → f$ uniformly and $g_n→g$ uniformly, then $f_n + g_n →f + g$ ...
0
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0answers
28 views

Uniform convergence on subintervals

(a) Fix a positive integer $M$ and let $\{f_{n} : [0, M]\rightarrow \mathbb{R}\}$ be a sequence of functions. Suppose that $f_{n}\rightarrow f$ pointwise on $[0, M]$ and that $f_{n}\rightarrow f$ ...
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0answers
19 views

Problem about Riemann integrable function's uniform convergence

I have no idea of how to answer the following question. It seems that the function is recurrence.
0
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3answers
27 views

Proving uniform convergence of a simple series

I have a math question that I desperately need help on. I need to prove that a series is uniform convergent on $(-1,1)$. The series is $$\sum_{k=1}^{\infty} k^2 x^k$$ I tried to use the $M$-test for ...
1
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1answer
43 views

A proof on uniform convergence of polynomials with bounded degree

In $[0,1]$ suppose that you have a sequence of polynomials $(P_n)_{n\in\mathbb{N}}$ of at most degree $M$ each. Also, suppose that $P_n(x) \rightarrow 0$ pointwise for every $x\in[0,1]$. Is is true ...
0
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3answers
34 views

Uniform convergence of $f_n(x) = n \sin(\frac{x}{n}) , x \in [-r,r]$

It is asked to prove that $$f_n(x) = n \sin(\frac{x}{n}) , x \in [-r,r]$$ Converges uniformly on the given interval for $r>0.$ The resolution of this suggested considered the fact that the ...
0
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4answers
29 views

Uniform convergence of $f_n(x) = 1 + x + … + x^n$

It is asked to prove that the sequence $$f_n(x) = 1 + x + ... + x^n$$ Uniformly converges in $[0, r] (0<r<1)$ to $$f(x)=\frac{1}{1-x}$$ That is, i need to show that $$\lim_{n \to \infty} || ...
2
votes
1answer
44 views

Uniform convergence

How would you show that $$f(x)=\sum_{n=2}^\infty \frac{\sin(2\pi n x)}{n\log n}$$ converges uniformly on $x\in[0,1]$. The pointwise convergence can be proved by Drichlet test
4
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2answers
57 views

Continuous functions satisfying a condition to be convex

Let $f$ be continuous on $\mathbb R$, and satisify $$f(x)\leq \frac{1}{2h}\int_{-h}^h f(x+t)d t, \forall\ h>0.$$ Show that $f$ is convex. The original question is "if and only if". However, I ...
0
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1answer
42 views

Uniform convergence of $f_n(x) = n \sin(\frac{\pi x}{n})$ in [a,b]

Show that the sequence of functions $$f_n(x) = n \sin \Bigl(\frac{\pi x}{n} \Bigr)$$ Converges uniformly in any interval of the form $[a,b]$ My attempt: The pointwise limit of this function is $$ ...
0
votes
2answers
67 views

Check the uniform convergence of $\sum\limits_{n = 1}^{\infty} \frac{n^2}{x} \exp(\frac{-n^2}{x})$

$$\sum\limits_{n = 1}^{\infty} \frac{n^2}{x} \exp{\frac{-n^2}{x}}$$ where $ 0 < x < \infty$ While $$\lim\limits_{n\rightarrow\infty} (a_n)^{1/n} = 0$$ The sum would converge. But how to check ...
0
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0answers
28 views

Uniform convergence of function

Consider $$f_C(x)=\sum_{n=2}^\infty \frac{\cos(2\pi nx)}{n\log n}$$ Show that it does not converges uniformly and $f_C(x)\geq c\log \log \frac{1}{|x|}$ as $x\rightarrow 0$. I used summation parts ...
2
votes
2answers
51 views

show a series is not convergence uniformly

I am trying to prove that the series $$ \sum_{n=1}^\infty f_n(x)=\sum_{n=1}^\infty-x\exp(-n^2 x^2)$$ is not convergence uniformly in $[-1,1]$. What I usually do when I trying to prove a series is ...
9
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2answers
155 views

Uniform convergence of the series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\sin\frac{1}{nx}$ on $(0,+\infty)$

Does the series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\sin\frac{1}{nx}$$ converge uniformly on $(0,+\infty)$? First, I guessed that it does, but Abel's test is not applicable here because ...
3
votes
3answers
47 views

The uniform convergence

Let $u_n$ be a decreasing sequence of continuous functions on a compact Hausdorff $X$ and assume $u_n(x)\to0$ for all $x\in X$.Then I want to prove $u_n\to0$ uniformly. I am trying to use ...
0
votes
0answers
27 views

'Real analysis' pointwise convergence

I have a question about real analysis. Determine whether $$\sum_{n=1}^\infty \frac{x+x^3(k-k^2)}{(1+k^2x^2)(1+(k-1)^2x^2)}$$ converges pointwise to some $S(x)$ and $S(x)$ is continuous. ($x \in ...
1
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1answer
36 views

Assuming uniqueness in Peano Theorem for ODEs, show that the sequence of approximate solutions is uniformly convergent.

If we know existence of solutions for $$ x'=f(t,x), \quad x(t_0)=x_0, $$ obtained by Peano's Theorem, and furthermore we know that the solution in unique ( we have not assumed satisfaction of ...
0
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0answers
43 views

Uniform convergence of series on [0,1]

Show that the series $(1-x)^2+x(1-x)^2+x^2(1-x)^2+\cdots$ is not uniformly convergent on $[0,1]$ My try: $S_n(x)=(1-x)(1-x^n)$ The pointwise sum $S(x)=1-x$ Now using the $M_n$ test ...
1
vote
1answer
67 views

Use Hurwitz or Schwartz?

Let $U\subset\mathbb C$ be a bounded region (i.e., open and connected) and $f$ be a holomorphic function on $U$, with $f(U)\subset U$. Denote by $f_{n} = f \circ f \circ · · · \circ f $. Suppose ...
0
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1answer
19 views

Need correction for my “proofs” about the integrable functions.

Let $(X,\mathcal{A},\mu)$ be a measureable space, and assume that $\mu(X)<\infty$. Let $\left \{ u_{n} \right \}_{n\geq 1}$ be a sequence of functions in $\mathcal{L}^{1}(\mu)$ that converges ...
1
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0answers
18 views

Showing equality between integral and shifted Fourier series

Let $f\in E[-\pi,\pi]$ and let $f\sim \frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx +b_n\sin nx$ be the fourier series of f in $[-\pi,\pi]$.show that $$\forall -\pi\le c,x\le \pi \quad ...
2
votes
1answer
33 views

Proving that $\int_0^t s(x) dx = \frac{t^2}{1-t}$

Let $ s =s(x)$ given by $s(x) = \sum_{k=1}^\infty (k+1)x^k. $ Prove that for all $ t \in ]-1,1[,$ $$\int_0^t s(x) dx = \frac{t^2}{1-t}$$ Conclude that, for all $x \in ]-1,1[,$ $$\sum_{k=1}^\infty ...
0
votes
0answers
21 views

Uniform and absolute convergence of $\sum \frac{2^n}{n^2+1} x^{2n}$

Find the radius of converence of the series $$\sum \frac{2^n}{n^2+1} x^{2n}$$ and analyze the absolute and/or uniform convergence My attempt The convergence radius of this serie is ...
0
votes
2answers
28 views

Uniform convergence on the interval of convergence

I am with a lot of doubts about the uniform convergence of a power series. For example, consider the series $$\sum \frac{1}{n} x^n$$ It is easy to find that the radius of convergence of this series ...
4
votes
1answer
107 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
1
vote
5answers
53 views

Showing uniform convergence

I am struggling to show that $f_n$ converges uniformly to $|f|$, where $\displaystyle{f_n=\frac{|f|^2}{|f|+\frac{1}{n}}}$. I have that ...
1
vote
2answers
26 views

$f_n$ converges uniformly on $\overline\Omega$

Suppose $\Omega$ be a bounded region and $\{f_n\}_{n\in\mathbb N}$ a sequence of continuous functions on $\overline\Omega$ which are holomorphic in $\Omega$ and $f_n$ converges uniformly on the ...
3
votes
1answer
84 views

Uniform convergence on an interval

Let $a<c<b$. Let {$f_n$} be a sequence of functions converging uniformly on $[a,c]$ and $[c,b]$. Prove that {$f_n$} converges uniformly on $[a,b]$ My attempt: Intuitively, I see that {$f_n$} ...
2
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1answer
35 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
0
votes
1answer
31 views

Uniform convergence of the series $\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $

I am trying to find if this series is uniformly convergent: $$\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $$ So far I have (using the Weierstrass M-Test): $$| \frac{\cos(2nt)}{4 n^2 - 1}| \le ...
3
votes
2answers
75 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
0
votes
1answer
34 views

Uniform convergence and relative error?

I never took a class where uniform convergence was introduced, so I only know the Definition and not much more about it. I have an Approximation of some sequence of functions $f_n$ by some function ...
1
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1answer
41 views

Convergence of a sum of sines

If $ s_N(x) := \sum_{n = 1}^N c_n \sin(n x) $ converges uniformly on $[0, \pi]$ as $N \to \infty$ then $c_n = o(n^{-1})$. a) Is $c_n = o(n^{-1})$ sufficient for uniform convergence? b) Is $\sum_n n ...
1
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1answer
28 views

Limit of series of function, uniform convergence?

$ f(x) = \frac{1}{x} $, $ g(x) = \sum_{n=1}^{\infty} \left( 2^{n-1} \sin \frac{x}{2^{n-1}} \sin^2 \frac{x}{2^n} \right)$ What is the limit of $ f(x)g(x) $ at $ x\to 0 $? Can I exchange lim and sigma ...
0
votes
0answers
17 views

A regulated function is the limit of a normally convergent series of step functions.

Let $a<b\in\mathbb{R}$ and $E$ a Banach space. A function $f:[a,b]\rightarrow E$ is called regulated if it has right-sided and left-sided limits at any point of $[a,b]$ or equivalently if it is the ...
2
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1answer
58 views

Uniform integrability of a sequence of functions.

I am considering the following sequence of functions. I think it converges pointwise to $0$ because the intervals in the domain in which the nth function is greater than zero eventually shrink and ...
0
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1answer
38 views

Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
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1answer
47 views

Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
2
votes
4answers
65 views

Series $\sum_{n=0}^{\infty}\frac{1}{2^n-1+e^x}$ properties

We need to prove $$f(x)=\sum_{n=0}^{\infty}\frac{1}{2^n-1+e^x}<\infty\ \ \forall_{x\in\mathbb{R}}$$ and then find all continuity points and all points in which $f$ is differentiable + calculate ...
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2answers
40 views

How to take the limit of the improper integral of a sequence of functions

Suppose $f_1, f_2, . . .$ are (Riemann) integrable functions. Then what is the $\epsilon$ definition of $$\lim_{n \rightarrow \infty} \lim_{M \rightarrow \infty} \int_{0}^{M} f_n(x) dx = L $$ for $L ...
1
vote
1answer
27 views

What kind of convergence is $\sum |f_n|$?

We know that $\sum |f_n|$ converges on $E\subseteq \mathbb R$ then $\sum f_n$ is said to converge absolutely on $E$. But in terms of pointwise or uniform convergence, I am willing to know what kind ...
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0answers
25 views

How many sequence of functions are there to converge pointwise to a given function on $E\subseteq \mathbb R$?

yesterday night, I was studying sequence of functions in $\mathbb R$ and then this question came to mind. When a sequence of real valued function is given, we can find out it's pointwise limit ...
2
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1answer
28 views

If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.

If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\} $ will converge uniformly to $fg$? I am absolutely stuck on this. Please help.
0
votes
0answers
42 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
3
votes
0answers
171 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
1
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0answers
15 views

Uniform convergence with respect to a parameter

So this is just a notational question. Assume one has a sequence $f_n\to f$ uniformly, where $f_n,f:X\to Y$ for some metric/Banach spaces $X,Y$. Now suppose that $f_n$ and $f$ depend on a parameter ...
2
votes
1answer
40 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
4
votes
1answer
32 views

Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that ...
2
votes
1answer
47 views

Complex Analysis Weierstrass M-Test

Prove that each of the following series converges uniformly on the corresponding subset of $\mathbb C$: $$\begin{align*} \text{(a)} \; & \sum_{n=1}^\infty \frac{1}{n^2 z^{2n}}, & & ...