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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1answer
21 views

Convergence of $f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1]$

$(f_n)$ is a succession of functions $$ f_n : [-1,1] \rightarrow \mathbb{R} \\ \\ f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1] $$ Punctual convergence $\forall x \in ...
0
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2answers
30 views

Uniform Convergence of $\sum_{i=1}^\infty \arctan\left(\frac{x}{i^2}\right)$ and its differentiabilty

I was trying to prove it is uniform convergent by it is Cauchy in sup-norm, since I don't know what does it converge to and it seems that M-test fail (as each term is bounded by $\pi/2$). ...
0
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1answer
30 views

Determine whether the following sequences (fn) converge uniformly, pointwise, or neither:

Determine whether the following sequences $(f_n) \in F(E, \mathbb{R})$ - where E is a set - converge uniformly, pointwise, or neither: a) $f_n(x) = \frac{n^2x} { 1 + n^2x^2}$ on set $E = \mathbb{R}$ ...
-1
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0answers
10 views

Uniformly convergent Laurent Series [on hold]

Why does a Laurent Series with positive and negative parts converge uniformly only on compact sets?
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1answer
26 views

Show ${f_n(x)}_ {n=1, \cdots, \infty}$ converges to $0$ uniformly on $(0,1)$

If $f_n(x) = \dfrac{x}{1+nx}$, show that ${f_n(x)}_{n=1, \cdots, \infty}$ converges uniformly to $0$ on $(0,1)$. Here is what I have so far: Let $\epsilon>0$ be given. Pick any $n \in ...
1
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3answers
35 views

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$. I do not know if that's so easy that I'm simply missing something, but I can't find any criterion which ...
0
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1answer
27 views

Uniform Convergence when $\lim f(x)$ does not exist

I can't find any similar questions online (except it might just be a bit too wordy), but say you are given a function: $f_n(x) = \frac{1}{1+x^n}, x \in \mathbb{R}$ And the limit of $f_n(x)$ does not ...
2
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1answer
26 views

Let the sequence of functions $f_n(x)$ equal $1$ if $x ∈ [n, n + 1)$ and $0$ otherwise. Why doesn't $f_n$ converge uniformly?

Let the sequence of functions $f_n(x)$ equal $1$ if $x ∈ [n, n + 1)$ and $0$ otherwise. How can I use the definition of uniform convergence to show that $f_n$ does not converges uniformly? If a ...
0
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2answers
42 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...
0
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0answers
23 views

Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to ...
1
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1answer
53 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
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3answers
52 views

Uniform convergence of $f_n(x)=\frac{1}{1+nx}$ on $(0,1)$

Consider the sequence of functions $f_n(x)=\frac{1}{1+nx}$ for $x\in (0,1)$. Then $f_n (x) → 0$ pointwise but not uniformly on $(0,1)$. $f_n (x) → 0$ uniformly on $(0,1)$. $\int_{0}^{1} f_n (x) dx ...
0
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1answer
24 views

Bounded Measure functions on a set of finite measure.

In Real Analysis of Royden and Fritzpatrick's book page 77, Proposition 8 proof first sentence states that " Since the convergence is uniform and each $f_n$ is bounded, the limit function f is ...
1
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1answer
23 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r ...
0
votes
2answers
26 views

Showing uniform convergence of a sequence

Let $f_n(x)= \frac x{x+n}$ for $n \in \Bbb{N}$. Show that if $a>0$ then $f_n$ converges to 0 uniformly on $[0,a]$ and show that the convergence is not uniform on $[0,\infty]$. So I've deduced that ...
-2
votes
0answers
17 views

Find all a $ \ge $ 0, that series convergence uniformly [closed]

Find all $a \ge 0$, that $f(x) = \sum\limits_{i=1}^n \frac{nsin(n)}{1+n^{3}x}$ convergence uniformly on $(a, \infty)$
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3answers
26 views

Continuity and uniform convergence counterexample

I have been having trouble finding a suitable counterexample to my problem, which I have written below. For each $n\geq 1$, let $f_{n}\colon \mathbb{R}\to\mathbb{R}$ be a continuous function and ...
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0answers
21 views

How to Visualize the uniform convergence of sequence and series of functions.

I've just started the "sequence and series of functions".I've read the definitions of uniform convergence & pointwise convergence,but somewhere I am getting problem in getting the intuition. It ...
0
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1answer
52 views

Examine convergence and (almost) uniform convergence of $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$.

How to examine convergence, almost uniform convergence and uniform convergence of series $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$?
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1answer
47 views

Convergence of the series $\sum_{n =1}^{\infty} (-1)^{n}x^{n}(1-x)$

Convergence of the series $w_n = (-1)^{n}x^{n}(1-x)$ on $(0,1)$, then $\sum w_n$ is uniformly convergent. Idea : I wanted to use the M-test for convergence of series,so I found out the value of x for ...
1
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1answer
63 views

Showing $\sum_{n=0}^{\infty} \frac {sin(nx)}n $ converges uniformly

I have to show that $$\sum_{n=0}^\infty \frac {\sin(nx)}n $$ converges uniformly on $[-\pi, -d]\cup[d, \pi]$ but not on $[-\pi, \pi]$ I can't really use Weierstrass's criterion, because the ...
1
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1answer
36 views

uniform convergence of the series $\sum_{n =1}^{\infty}e^{-nx} sin (nx) $

Prove that the series $\sum_{n =1}^{\infty}e^{-nx} sin (nx) $ for $x > 0$ is uniformly convergent. Idea: It satisfies the neccessary condition as $a_n \rightarrow 0$. Now use M-test for series ...
0
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0answers
31 views

Is $ f_n(x) = \frac{1}{1+(x-n)^2}$ uniformly convergent on $\mathbb R$?

Is $ f_n(x) = \frac{1}{1+(x-n)^2}$ uniformly convergent on $\mathbb R$? I think it isn't, because for $x=n$ we have $f_n(n) = 1$, and $f_n(x) \rightarrow 0$, so the uniform convergence inequality $$ ...
1
vote
1answer
16 views

Uniform convergence of function series, convergence on intervals

I I'm interested in, if it is true: If series $\sum f_k$ is uniformly convergence on every interval $[n; n+10]$, where $n \in \mathbb{Z}$, then $\sum f_k$ is uniformly convergence on $\mathbb{R}$. ...
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0answers
26 views

understanding a proof for uniform convergence of Fourier features

See part of the proof. In the proof, $\tilde{f} := \tilde{z}(x)'\tilde{z}(y) - k(x,y)$, the vector product is the approximation of a shift-invariant kernek $k(x,y)$. What is not clear to me is the ...
0
votes
1answer
25 views

$f_1(x)=sin(x),\; f_{n+1}(x)=sin(f_n(x))$, $(f_n)$ converges to zero uniformly

Consider for $n\in\mathbb{N}$ the function $f_n:\mathbb{R}\to\mathbb{R}$ given by $$f_1(x)=sin(x),\; f_{n+1}(x)=sin(f_n(x)).$$ I'm stuck to prove that $(f_n)_n$ converges to $f=0$ uniformly. First ...
0
votes
1answer
19 views

show that the following sequence function converges uniformly to 0 on the given set $\left\{\frac{\sin nx}{nx}\right\}$ on $[\alpha,\infty)$ …

... where $\alpha>0$. question: show that the following sequence function converges uniformly to $0$ on the given set $\displaystyle\left\{\frac{\sin nx}{nx}\right\}$ on $[\alpha,\infty)$ where ...
0
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2answers
49 views

A tricky question involving convergence of series

I'm struggling to find a title for this. Hopefully it can be edited subsequently. Suppose $\sum_k s_k(x)$ converges to $S(x)$ Now suppose each $s_k(x)$ has a sequence $t_k^N(x)$ approaching it (as ...
-1
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0answers
31 views

Example where uniform convergence does not imply normal convergence

So I would like to show that uniform convergence does not imply normal convergence and wanted to use the following example: $\sum_{i=2}^\infty g_n(x)$, where ...
2
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4answers
46 views

Proving uniform convergence of $(1+\frac{x}{n})^n$ to $e^x$ on compact intervals in the real numbers

My goal is to prove that if $b> a > 0$ are real numbers, then: $\lim_{n \rightarrow \infty} \int_a^b (1 + x/n)^n e^{-x} dx = b-a$. I think the best way to do this is to show that $(1+x/n)^n$ ...
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0answers
25 views

Help showing $u \in W_0^{1,p}(I)$ if and only if $u=0$ on $\partial I$

I am reading the proof the following statement provided in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by haim Brezis: If $u \in W_0^{1,p}(I)$, then $u=0$ on $\partial ...
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2answers
51 views

Normal convergence implies uniform absolute convergence but not the other way round

How do I show that normal convergence of a series implies uniform and absolute convergence? So, a series $f_1+f_2+...$ of functions $f_n:D\rightarrow\mathbb{C}, D\subset\mathbb{C}$ is normally ...
0
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1answer
18 views

${f_n}$ differentiable and $f'_n$ converges uniformly on $[a,b]$. How to show $f_n(x) - f_n(a)$ also converges?

I tried using 2 mean value theorems but I got 2 different x values: $(x-a)f_n'(c)$ and $(x-a)f_m'(d)$ so I couldn't make use of $f'_n$ unif convergence. What should I tweak?
1
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1answer
34 views

Find Maclaurin series for integral of $e^{z^2}$

I need to find a Taylor Series expansion of $\displaystyle \int_{0}^{z}e^{\zeta^{2}}d\zeta$ around $z=0$, which shouldn't be hard enough. Except that I can only integrate term-by-term if the Taylor ...
1
vote
1answer
67 views

Have I done this correctly?

I need to show that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n+x^2}$ converges uniformly but not absolutely on $\mathbb{R}$. First, I showed that the absolute value of the partial sums diverges for ...
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0answers
19 views

Interchanging Limit and Integral sign

I'm reading a book on composition operators, and the author makes the following claim: Given a self-map of the unit disc, and a $H^2$ function $f$, where $H^2$ is the Hardy space, if we fix a radius ...
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2answers
24 views

Uniform convergence on compact intervals of R [closed]

Does the sequence $f_n(x)=e^x(1+x/n)$ converge uniformly on compact intervals of R?
0
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1answer
38 views

Does $(1+x/n)e^x$ uniformly converge in $\mathbb{R}$?

Does the sequence $f_n(x)=e^x(1+x/n)$ converge uniformly on R? What would be the function it converges to if it does?
1
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3answers
56 views

Prove uniform convergence of a series

Define $g(x)$ = $\sum_{n=0}^\infty \frac{x^{2n}}{1+x^{2n}}$. Find the values of x where the series converges and show that we get a continuous function on this set. So I know that if I can prove ...
2
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0answers
14 views

uniform convergence empirical CDF depending on distributional parameters

Let $X$ be a (assume 1D) r.v. with density function $f_\theta$ (assume $\theta \in \Theta$, being $\Theta$ a compact set), where the parameter $\theta$ characterizes the distribution of $X$. The ...
0
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0answers
25 views

Local Uniform Convergence and Composition

I've been sitting down can't quite tell if this is true or not, but I suspect that it should be. Edit: Suppose that $\Omega$ is a open, connected subset of $\mathbb{C}$, and suppose that $(f_n) ...
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1answer
29 views

Is the series $\sum_{n=1}^\infty \dfrac {z^n}{1+z^n}$ of complex numbers uniformly convergent in $B(0,1)$?

Is the series $\sum_{n=1}^\infty \dfrac {z^n}{1+z^n}$ of complex numbers uniformly convergent in $B(0,1)$ ? I know it is point-wise convergent in $B(0,1)$ , I also know that it is uniformly convergent ...
1
vote
1answer
33 views

Under what conditions will $f(x)^n$ converge pointwise and uniformly?

Let $f(x)$ be a continuous function on $[0,1]$. Under what conditions on $f$ will the sequence $f_n(x)$ = $(f(x))^n$ converge uniformly and pointwise?
1
vote
1answer
23 views

Uniform convergence of the sequence $nx/e^{nx}$

Consider the sequence of functions $f_n(x) = nxe^{-nx}$. We notice that if $x < 0$ the sequence diverges to $-\infty$ while if $x \geq 0$ the sequence converges to $0$. This is the analysis of the ...
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0answers
41 views

Weierstrass M-test counterexample

Let's consider the uniform convergence of the integral $ \displaystyle \int\limits_0^\infty f(x,y)dx $ on the some segment $[c,d]$. We know the Weierstrass M-test as the sufficient condition of ...
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votes
2answers
37 views

Does this sequence of functions converge uniformly to $0$?

Suppose that $f$ is continuous and $|f(x)|<1$ for all $x$ in $[a,b]$. Does $[f(t)]^n$ converge uniformly to $0$? Is it a no? Because if I have $x^n$ then at $x$ very close to $1$ it takes very ...
0
votes
2answers
31 views

Proof of non-uniform convergence of $x^n$

The wikipedia page on uniform convergence indicates that $f_{n}:[0,1]\to[0,1]$ with $f_{n}(x):=x^{n}$ converges pointwise but not uniformly to $$ f(x)= \begin{cases} 0,\quad x \in [0,1)\\ 1,\quad ...
0
votes
1answer
18 views

General form for complex limit function $\sum p(n) z^n$ where $p \in \mathbb{C} [n]$

Given a polynomial $p \in \mathbb{C} [n]$ of degree $k$, I need to show that the power series $\sum_{n=1}^{\infty} p(n) z^n$ uniformly converges in the open unit disc, and that the limit function $f$ ...
0
votes
1answer
48 views

Pointwise and uniform convergence of a series of functions

Define a sequence of functions on $[0,\infty)$ such that $\forall n\in\mathbb{N}$, $$ f_n(x)\triangleq \begin{cases} 1 & x\in[n,n+\frac{1}{n}]\\ 0 & \text{otherwise} \end{cases} $$ Does the ...
1
vote
3answers
49 views

Pointwise convergence and uniform convergence converge to same limit?

Sometimes I see in proofs they say "this function is pointwise convergent to limit a. Let's check if it's uniformly convergent to limit a." And then they do a uniform convergence check on a and prove ...