Uniform convergence Definition and 164 Threads
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions
(
f
n
)
{\displaystyle (f_{n})}
converges uniformly to a limiting function
f
{\displaystyle f}
on a set
E
{\displaystyle E}
if, given any arbitrarily small positive number
ϵ
{\displaystyle \epsilon }
, a number
N
{\displaystyle N}
can be found such that each of the functions
f
N
,
f
N
+
1
,
f
N
+
2
,
…
{\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }
differ from
f
{\displaystyle f}
by no more than
ϵ
{\displaystyle \epsilon }
at every point
x
{\displaystyle x}
in
E
{\displaystyle E}
. Described in an informal way, if
f
n
{\displaystyle f_{n}}
converges to
f
{\displaystyle f}
uniformly, then the rate at which
f
n
(
x
)
{\displaystyle f_{n}(x)}
approaches
f
(
x
)
{\displaystyle f(x)}
is "uniform" throughout its domain in the following sense: in order to guarantee that
f
n
(
x
)
{\displaystyle f_{n}(x)}
falls within a certain distance
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
, we do not need to know the value of
x
∈
E
{\displaystyle x\in E}
in question — there can be found a single value of
N
=
N
(
ϵ
)
{\displaystyle N=N(\epsilon )}
independent of
x
{\displaystyle x}
, such that choosing
n
≥
N
{\displaystyle n\geq N}
will ensure that
f
n
(
x
)
{\displaystyle f_{n}(x)}
is within
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
for all
x
∈
E
{\displaystyle x\in E}
. In contrast, pointwise convergence of
f
n
{\displaystyle f_{n}}
to
f
{\displaystyle f}
merely guarantees that for any
x
∈
E
{\displaystyle x\in E}
given in advance, we can find
N
=
N
(
ϵ
,
x
)
{\displaystyle N=N(\epsilon ,x)}
(
N
{\displaystyle N}
can depend on the value of
x
{\displaystyle x}
) so that, for that particular
x
{\displaystyle x}
,
f
n
(
x
)
{\displaystyle f_{n}(x)}
falls within
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
whenever
n
≥
N
{\displaystyle n\geq N}
.
The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions
f
n
{\displaystyle f_{n}}
, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit
f
{\displaystyle f}
if the convergence is uniform, but not necessarily if the convergence is not uniform.
View More On Wikipedia.org
(
f
n
)
{\displaystyle (f_{n})}
converges uniformly to a limiting function
f
{\displaystyle f}
on a set
E
{\displaystyle E}
if, given any arbitrarily small positive number
ϵ
{\displaystyle \epsilon }
, a number
N
{\displaystyle N}
can be found such that each of the functions
f
N
,
f
N
+
1
,
f
N
+
2
,
…
{\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }
differ from
f
{\displaystyle f}
by no more than
ϵ
{\displaystyle \epsilon }
at every point
x
{\displaystyle x}
in
E
{\displaystyle E}
. Described in an informal way, if
f
n
{\displaystyle f_{n}}
converges to
f
{\displaystyle f}
uniformly, then the rate at which
f
n
(
x
)
{\displaystyle f_{n}(x)}
approaches
f
(
x
)
{\displaystyle f(x)}
is "uniform" throughout its domain in the following sense: in order to guarantee that
f
n
(
x
)
{\displaystyle f_{n}(x)}
falls within a certain distance
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
, we do not need to know the value of
x
∈
E
{\displaystyle x\in E}
in question — there can be found a single value of
N
=
N
(
ϵ
)
{\displaystyle N=N(\epsilon )}
independent of
x
{\displaystyle x}
, such that choosing
n
≥
N
{\displaystyle n\geq N}
will ensure that
f
n
(
x
)
{\displaystyle f_{n}(x)}
is within
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
for all
x
∈
E
{\displaystyle x\in E}
. In contrast, pointwise convergence of
f
n
{\displaystyle f_{n}}
to
f
{\displaystyle f}
merely guarantees that for any
x
∈
E
{\displaystyle x\in E}
given in advance, we can find
N
=
N
(
ϵ
,
x
)
{\displaystyle N=N(\epsilon ,x)}
(
N
{\displaystyle N}
can depend on the value of
x
{\displaystyle x}
) so that, for that particular
x
{\displaystyle x}
,
f
n
(
x
)
{\displaystyle f_{n}(x)}
falls within
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
whenever
n
≥
N
{\displaystyle n\geq N}
.
The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions
f
n
{\displaystyle f_{n}}
, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit
f
{\displaystyle f}
if the convergence is uniform, but not necessarily if the convergence is not uniform.
View More On Wikipedia.org
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I Uniform convergence and derivatives -- difference between two theorems?
The first theorem is from here (page 9 in the pdf): The second theorem is from baby Rudin: Obviously the open/closed intervals is one difference, but if we were to replace the open intervals with closed intervals, is the first theorem a special case of the second one? Some have said they are...- psie
- Thread
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- Tags
- Uniform convergence
- Replies: 1
- Forum: Calculus
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I Switching a derivative and a limit is sometimes possible, sometimes not...
Here is a cute calculation about which I have my doubts: Treating the derivative as a limit makes the first step a case of switching the order of limits. One cannot automatically do this, as for example for the sequence of functions: More precisely, that one should be able to switch limits iff... -
M
I Question about uniform convergence in a proof
The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265) Proposition 14.5. Suppose ##D## is a bounded simply connected open set in the plane, and let ##\phi: D \rightarrow \mathbb{D}## be a conformal equivalence. (i) If ##\zeta## is a simple boundary point of...- MathLearner123
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- Forum: Calculus
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Z
Prove that this series converges uniformly on R
Here is the solution I came up with Consider the sequence of functions ##\{f_n\}=\left \{ \frac{x}{n(1+nx^2)}\right \}## defined on ##\mathbb{R}##. By differentiating ##f_n(x)## and equating to zero we find critical points at ##x=\pm \frac{1}{\sqrt{n}}##. By checking the second derivative we...- zenterix
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- Tags
- Uniform convergence
- Replies: 4
- Forum: Calculus and Beyond Homework Help
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Show function series involving arctan is not differentiable at x=0
I have previously shown that the function series is differentiable at ##x\neq 0##. The series converges uniformly (thus pointwise) on ##\mathbb R## and the term wise differentiated series is uniformly convergent on any interval ##d\leq |x|##, where ##d>0##. Moreover, the terms are continuously...- schniefen
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- Replies: 26
- Forum: Calculus and Beyond Homework Help
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MHB Uniform convergence - Length of graph
Hey! :giggle: We define the sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ by $$f_{n+1}(x)=\begin{cases}0 & \text{ if } x\in \left[ 0, \frac{1}{2n+3}\right ]\\ |2(n+1)x-1| & \text{ if } x\in \left [\frac{1}{2n+3}, \frac{1}{2n+1}\right ] \\ f_n(x) & \text{ if } x\in \left...- mathmari
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- Replies: 21
- Forum: Topology and Analysis
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MHB Sequence of functions : pointwise & uniform convergence
Hey! 😊 Let $0<\alpha \in \mathbb{R}$ and $(f_n)_n$ be a sequence of functions defined on $[0, +\infty)$ by: \begin{equation*}f_n(x)=n^{\alpha}xe^{-nx}\end{equation*} - Show that $(f_n)$ converges pointwise on $[0,+\infty)$. For an integer $m>a$ we have that \begin{equation*}0 \leq...- mathmari
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- Replies: 11
- Forum: Topology and Analysis
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Uniform convergence of a sine series
I'm not too sure how to use the hint here. What I had so far was this: an odd extension of ##f## implies ##f = \sum_{k=1}^\infty b_k \sin(k x)##. Notice for ##m>n## $$ \left|\sum_{k=1}^m b_k\sin(k x) - \sum_{k=1}^n b_k\sin(k x)\right| = \left| \sum_{k=n+1}^m b_k\sin(k x)\right| \leq...- member 428835
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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Uniform convergence of a sequence of functions
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...- AllRelative
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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Showing Uniform Convergence of Cauchy Sequence of Functions
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 >...- fishturtle1
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
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Interval of uniform convergence of a series
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...- NicolaiTheDane
- Thread
- Replies: 24
- Forum: Calculus and Beyond Homework Help
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Uniform convergence of a parameter-dependent integral
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...- Peter Alexander
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
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I Convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##
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... -
Pointwise vs. uniform convergence
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...- Rectifier
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- Forum: Calculus and Beyond Homework Help
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I If A is an algebra, then its uniform closure is an algebra.
Let me give some context. Let X be a compact metric space and ##C(X)## be the set of all continuous functions ##X \to \mathbb{R}##, equipped with the uniform norm, i.e. the norm defined by ##\Vert f \Vert = \sup_{x \in X} |f(x)|## Note that this is well defined by compactness. Then, for a...- member 587159
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- Replies: 4
- Forum: Topology and Analysis
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I Pointwise and Uniform Convergence
Hello! Can someone explain to me in an intuitive way the difference between pointwise and uniform convergence of a series of complex functions ##f_n(z)##? Form what I understand, the difference is that when choosing an "N" such that for all ##n \ge N## something is less than ##\epsilon##, in the...- Silviu
- Thread
- Replies: 5
- Forum: Topology and Analysis
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MHB Power series and uniform convergence.
Hi. I have this power serie (2^n/n)*z^n that runs from n=1 to infinity, and I have to show whether it's uniform konvergence on [-1/3, 1/3] or not. I hope someone can help me with this.- MissC
- Thread
- Replies: 2
- Forum: Topology and Analysis
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MHB Pointwise and uniform convergence
Hey! :o I want to check the pointwise and uniform convergence for the following sequences or series of functions: $f_n:[0, \infty)\rightarrow \mathbb{R}, f_n(x)=xe^{-nx}$ for all $n\in \mathbb{N}$ $f_n:[0, \infty)\rightarrow \mathbb{R}, f_n(x)=nxe^{-nx}$ for all $n\in \mathbb{N}$...- mathmari
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- Forum: Linear and Abstract Algebra
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I What do I need to know to understand Uniform convergence?
Hi, I started to study the function of Weierstrass (https://en.wikipedia.org/wiki/Weierstrass_function) And in one part says that the sum of continuous functions is a continuous function. i understand this but the Limiting case is a different history depend of the convergence, so what i need...- MAGNIBORO
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- Forum: General Math
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Punctual and uniform convergence
Homework Statement ##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##. We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if $$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x...- Kernul
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- Forum: Calculus and Beyond Homework Help
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I Limits of multivariable functions (uniform convergence)
Hi everybody! I'm preparing an exam of "Analysis II" (that's how the subject's called in German), and I have trouble understanding how to find the limit of a multivariable function, especially when it comes to proving the uniform convergence. Here is an example given in the script of my teacher...- JulienB
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- Forum: Topology and Analysis
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Pointwise, uniform convergence of fourier series
Hello; I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand Homework Statement Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems...- lilphy
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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Understanding convergence in norm, uniform convergence
Homework Statement Find an example of a sequence ##\{ f_n \}## in ##L^2(0,\infty)## such that ##f_n\to 0 ## uniformly but ##f_n \nrightarrow 0## in norm. Homework Equations As I understand it we have norm convergence if ##||f_n-f|| \to 0## as ##n\to \infty## and uniform convergence if there...- Incand
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MHB Range of Uniform convergence of dirchlet series
Find the Range of Uniform convergence of $ \zeta\left(x\right) = \sum_{n=1}^{\infty}\frac{1}{{n}^{x}} $ Using the Weierstrass-M test, I get this converges for $ 1 \lt x \lt \infty $ But the book's answer is $ 1 \lt s \le x \lt \infty $? I have scoured the book but can't see why they say it... -
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MHB Uniform convergence of a complex power series on a compact set
I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...- kalish1
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- Forum: Topology and Analysis
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Continuity and Differentiability of Infinite Series
Homework Statement I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)} The problem had three parts. The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2) The second was to prove that the function f(x) was continuous...- AnalysisNewb
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- Forum: Calculus and Beyond Homework Help
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Uniform Convergence: Understanding the Limitations and Implications
So I'm reading "An Introduction to Wavelet Analysis" by David F. Walnut and it's saying that the following sequence " (x^n)_{n\in \mathbb{N}} converges uniformly to zero on [-\alpha, \alpha] for all 0 < \alpha < 1 but does not converge uniformly to zero on (-1, 1) " My problem is that isn't...- BrainHurts
- Thread
- Replies: 12
- Forum: Topology and Analysis
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MHB Why Does Choosing z=1/n Demonstrate Non-Uniform Convergence?
on page 4, example 9 in this link, http://www.personal.psu.edu/auw4/M401-notes1.pdf, they show a sequence of functions is not uniformly convergent. To show this, you need to show that for some epsilon, there is no 'universal' N. But they didn't pick a particular value of $z$, they chose...- Fermat1
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- Replies: 4
- Forum: Topology and Analysis
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Uniform Convergence of a Sequence of Functions
Homework Statement Define f_n : \mathbb{R} \rightarrow \mathbb{R} by f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}} Show that f_n(x) \rightarrow |x| converges uniformly on compact subsets of \mathbb{R} Show that the convergence is uniform in all of \mathbb{R}...- BrainHurts
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- Replies: 14
- Forum: Calculus and Beyond Homework Help
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Proof uniform convergence -> continuity: Why use hyperhyperreals?
A uniformly convergent sequence of continuous functions converges to a continuous function. I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I... -
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Range of uniform convergence for a series
Homework Statement Find the range of uniform convergence for the following series η(x) = ∑(-1)n-1/nx ζ(x) = ∑1/nx with n ranging from n=1 to n=∞ for both Homework Equations To be honest I'm stumped with where to begin altogether. In my text, I'm given the criteria for uniform...- leonardthecow
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- Replies: 24
- Forum: Calculus and Beyond Homework Help
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MHB Pointwise convergence implies uniform convergence
Hi, I have to prove the following theorem: Let $f_n:[0,1] \to \mathbb{R}, \forall n \geq 1$ and suppose that $\{f_n|n \in \mathbb{N}\}$ is equicontinuous. If $f_n \to f$ pointwise then $f_n \to f$ uniformly. Before I start the proof I'll put the definitions here: $f_n \to f$ pointwise if and...- Siron
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- Replies: 1
- Forum: Topology and Analysis
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Proving Non-Uniform Convergence: Understanding the Role of Singular Points
I have a question where I am supposed to show that a series does not converge uniformly, I get the majority of the question, but one part in the solution I can't see the rationale or how they decided on the result: It has to do with the partial sum: SN= (1 - (-x2)N+1)/ (1+x2) The...- trap101
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- Replies: 6
- Forum: Calculus and Beyond Homework Help
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Uniform Convergence Homework: Is ##f_n(x) = \frac{x}{1+nx^2}##?
Homework Statement Is the sequence of function ##f_1, f_2,f_3,\ldots## on ##[0,1]## uniformly convergent if ##f_n(x) = \frac{x}{1+nx^2}##? 2. The attempt at a solution I got the following but I think I did it wrong. For ##f_n(x) = \frac{x}{1+nx^2}##, I got if ##f_n \to0## then we must...- Lee33
- Thread
- Replies: 4
- Forum: Calculus and Beyond Homework Help
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Uniform Convergence of ##\{f_n\}## on ##[0,a]##
Homework Statement Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.2. The attempt at a solution Is this correct? Should I add more detail? Thanks for your help! Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to...- Lee33
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
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Uniform convergence of sequence of functions
Homework Statement Let f_{n}(x)=\frac{x}{1+x^n} for x \in [0,∞) and n \in N. Find the pointwise limit f of this sequence on the given interval and show that (f_{n}) does not uniformly converge to f on the given interval. Homework Equations The Attempt at a Solution I found that the pointwise...- phosgene
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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E
Uniform convergence of a product of functions
Homework Statement Let \left[a,b\right] be a closed bounded interval, f : [a,b] \rightarrow \textbf{R} be bounded, and let g : [a,b] \rightarrow \textbf{R} be continuous with g\left(a\right)=g\left(b\right)=0. Let f_{n} be a uniformly bounded sequence of functions on \left[a,b\right]. Prove...- epr1990
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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Deceptive uniform convergence question
Homework Statement http://gyazo.com/55eaace8994d246974ef750ebeb36069 Homework Equations Theorem III : http://gyazo.com/af2dfeb33d3382430d39f275268c15b1 The Attempt at a Solution At first this question had me jumping to a wrong conclusion. Upon closer inspection I see the...- STEMucator
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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Proving Uniform Convergence of ##s_n(x)## to ##s(x)## on ##[b, ∞)##
Homework Statement Suppose that ##s_n(x)## converges uniformly to ##s(x)## on ##[b, ∞)##. If ##lim_{x→∞} s_n(x) = a_n## for each n and ##lim_{n→∞} a_n = a## prove that : ##lim_{x→∞} s(x) = a## Homework Equations ##\space ε/N## The Attempt at a Solution I see a quick way to do this one...- STEMucator
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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Uniform convergence for heat kernel on unit circle
Homework Statement I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow. H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . Homework Equations The Attempt at a...- stripes
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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MHB Families of holomorphic functions and uniform convergence on compact sets
Consider the sequence $\{f_n\}$ of complex valued functions, where $f_n=tan(nz)$, $n=1,2,3\ldots$ and $z$ is in the upper half plane $Im(z)>0$. I want to show two facts about this sequence: 1) it's uniformly locally bounded: for every $z_0=x_0+iy_0$ in the upper half plane, ther exist...- pantboio
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- Replies: 1
- Forum: Topology and Analysis
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J
Uniform Convergence and the Uniform Metric
Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should...- jmjlt88
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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A
A question about uniform convergence
Homework Statement For question 25.15 in this link: http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw9sum06.pdf I have some questions about pointwise convergence and uniform convergence... Homework Equations The Attempt at a Solution Our textbook says...- Artusartos
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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S
Uniform convergence integration
f is a continuous function on [0,infinity) such that 0<=f(x)<=Cx^(-1-p) where C,p >0 f_k(x) = kf(kx) I want to show that lim k->infinity ∫from 0 to 1 of f_k(x) dx exists so my idea is if I have that f_k(x) converges to f(x)=0 uniformly which I was able to show and that f_k(x) are all...- samithie
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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S
Uniform Convergence: Intuitive Explanation
I'm wondering about uniform convergence. We're looking at it in my complex analysis class. We are using uniform convergence of a series of functions, to say that we can interchange integration of the sum, that is: \int\sum b_{j}z^{j}dz=\sum\int b_{j}z^{j}dz=\int f(z)dz On an intuitive level I...- stephenkeiths
- Thread
- Replies: 2
- Forum: Topology and Analysis
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S
Prove Uniform Convergence of f_n=sin(z/n) to 0
Homework Statement I need to show that f_{n}=sin(\frac{z}{n}) converges uniformly to 0. Homework Equations So I need to find K(\epsilon) such that \foralln \geq K |sin(\frac{z}{n})|<\epsilon I'm trying to prove this in an annulus: \alpha\leq |z| \leq\beta The Attempt at a Solution I'm having...- stephenkeiths
- Thread
- Replies: 4
- Forum: Calculus and Beyond Homework Help
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M
Uniform Convergence of Power Series
Given a power series \sum a_n x^n with radius of convergence R, it seems that the series converges uniformly on any compact set contained in the disc of radius R. This might be a silly question, but what's an example of a power series that doesn't actually also converge uniformly on the whole...- markiv
- Thread
- Replies: 6
- Forum: Topology and Analysis
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Uniform Convergence of Poisson Kernel on [-π, π] minus (-a, a)
Homework Statement show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a) Homework Equations the integral of poisson...- hedipaldi
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- Forum: Calculus and Beyond Homework Help
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Uniform convergence and derivatives question
In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}. What I don't like about the theorem is the huge amount of assumptions required: " Suppose that {fn} is a sequence of functions which are differentiable on... -
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Uniform Convergence of Fourier sine and cosine series
Homework Statement f(x)= {1, ‐1/2<x≤1/2} {0, ‐1<x≤ ‐1/2 or 1/2<x≤1} State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞ Homework Equations The Attempt at a Solution...- hwill205
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- Forum: Calculus and Beyond Homework Help