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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N
Need to find if a sequence of functions has uniform convergence
Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...- natasha d
- Thread
- Replies: 13
- Forum: Calculus and Beyond Homework Help
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Δ
Uniform convergence of a quotient
Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...- δοτ
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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P
MHB Testing Uniform Convergence of Complex Function Sequences with Natural Numbers
How do I determine whether the following sequences of complex functions converge uniformly? i) z/n ii)1/nz iii)nz^2/(z+3in) where n is natural number- Poirot1
- Thread
- Replies: 1
- Forum: Topology and Analysis
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MHB Uniform Convergence of $Y(x)$ in $(0,1]$
Hello!A little problem:With the given series,$$Y(x)= \sum_{n=1}^{\infty}(-1)^n\frac{x^n\ln^nx}{n!} $$ ,why $Y(x)$ is Uniformly converges for all $x\in(0,1]$ ?Ok, I know that $Y(x)$ is u.c by M-test: $$\max{|x\ln{x}|}=\frac{1}{e}$$ And, $$ \sum_{n=0}^{\infty}\frac{(\frac{1}{e})^n}{n!} $$ Is...- Also sprach Zarathustra
- Thread
- Replies: 5
- Forum: Topology and Analysis
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Check Uniform Convergence of f_n(x): Solutions & Explanations
Homework Statement f(x) is defined within [a,b]. f_n(x)=\frac{\big\lfloor nf(x) \big\rfloor}{n} Check if f_n(x) is uniform convergent. The Attempt at a Solution This one seems to be easy however since I didn't touch calculus for quite a time I'm not confident with my solution. |\frac...- estro
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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A
Trying to Prove Uniform Convergence: Analysis II
Homework Statement I have a solution to the following problem. I feel it is somewhat questionable though If fn converges uniformly to f, i.e. fn\rightarrowf as n\rightarrow∞ and gn converges uniformly to g, i.e. gn\rightarrowf as n\rightarrow∞ , Prove that fngn...- Abraham
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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M
How Can I Prove Uniform Convergence of This Function as ρ→0?
Probably is a silly question, but how could I prove that the function (expressed in polar coordinates) \left(\rho^4\cos^2{\theta} + \sin^3{\theta}\right)^{\frac{1}{3}} - \sin{\theta} converges to 0 as rho->0 uniformely in theta (if it is true, of course)? -
H
Is the Function Uniformly Convergent on (0,1]?
I am given f_n(x)=\frac{nx}{nx+1} defined on [0,\infty) and I have that the function converges pointwise to 0 \ \mbox{if x=0 and} 1\ \mbox{otherwise} Is the function uniform convergent on [0,1] ? No. If we take x=1/n then Limit_{n\rightarrow\infty}|\frac{1/n*n}{1+1/n*n}-1|=0.5... -
Z
Uniform Convergence of Fourier Series
Homework Statement Find the minimum number required (value of n) for the average deviation of the Fourier Series to fall below 2% Homework Equations Use the Uniform Convergence of Fourier Series. Where Sm is the partial sum of the Fourier Series. C is constant. Here C is ∏^2 So...- ZedCar
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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T
Product of two sequences of functions [uniform convergence]
Homework Statement This is a homework question for a introductory course in analysis. given that a) the partial sums of f_n are uniformly bounded, b) g_1 \geq g_2 \geq ... \geq 0, c) g_n \rightarrow 0 uniformly, prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...- timon
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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M
Two real analysis problems: proving constancy and a uniform convergence problem
The problem statement Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b]. Show that f(x)=0,x\in [a,b]. The attempt at a solution It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...- Mr.Miyagi
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
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K
Cauchy Sequence of Continuous Fns: Uniform Convergence?
Is it true that a cauchy sequence of continuous functions defined on the whole real line converges uniformly to a continuous function? I thought this was only true for functions defined on a compact subset of the real line. Am I wrong? -
L
Uniform convergence, mean convergence, mean-square convergence
Hi, I was always troubled by the relationships between these modes of convergence (L^1, L^2, and L^{\infty} convergences, to be precise), so I took some books and decided to establish some relations between them. For some, I succeeded, for others I did not. Here's what I did so far: If I... -
D
Showing the uniform convergence of a gaussian function-like series
Homework Statement Prove that the series \sum_{n=0}^\infty e^{-n^2x^2} converges uniformly on the set \mathbb{R}\backslash\ \big] -\epsilon,\epsilon\big[ where \epsilon>0Homework Equations n/aThe Attempt at a Solution I have tried using Weierstrass M-test but I have not been able to find a...- dane502
- Thread
- Replies: 9
- Forum: Calculus and Beyond Homework Help
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L
Uniform Convergence of Sequences
Homework Statement For each of the following sequences (fn), find the function f such that fn --> f. Also state whether the convergence is uniform or not and give a reason for your answer. Homework Equations a.) fn(x) = 1/xn for x greater than or equal to 1 b.) f[SUB]n[SUB](x) =...- LauraLovies
- Thread
- Replies: 5
- Forum: Calculus and Beyond Homework Help
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B
Uniform convergence ( understanding how to apply a theorem)
Homework Statement show a function f_n is not uniformly convergent using a theorem: Homework Equations if f_n converges uniformly to F on D and if each f_n is cont. on D, then F is cont. on D The Attempt at a Solution not really sure what to do. use the contrapositive? would that...- bennyska
- Thread
- Replies: 11
- Forum: Calculus and Beyond Homework Help
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M
Uniform Convergence of g_n (x): Proof & Subset Analysis
1. Let g_n (x)=nx*exp(-nx). Is the convergence uniform on [0, ∞)? On what subsets of [0,∞) is the convergence uniform? 3. I am looking for a proof of how the convergence is uniform (possibly using Weierstrass' M Test?). I understand that the subset that determines uniform convergence is...- mscudder3
- Thread
- Replies: 18
- Forum: Calculus and Beyond Homework Help
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S
Basic Complex Analysis: Uniform convergence of derivatives to 0
Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...- snipez90
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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M
Distribution Theory - Uniform Convergence
F: C(Omega) -> D'(Omega); F(f) = F_f -- O = Omega Introduce the notion of convergence on C(Omega) by f_p -> f as p -> inf in C(O) if f_p(x) -> f(x) for any xEO Show that then F is a continuous map from C(O) to D'(O) Hint: Use that if a sequence of continuous functions converges to a...- MidnightR
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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J
Pointwise and uniform convergence of sequence of functions
Homework Statement show if has a uniform convergence of pointwise also we know that x gets values from 0 to 1The Attempt at a Solution for the pointwise I think its easy to show that limfn(x) as n->infinity is 0 but I am really stuck in uniform convergence I know that fn converges...- Jncik
- Thread
- Replies: 8
- Forum: Calculus and Beyond Homework Help
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H
Stone-Weierstrass, uniform convergence
Homework Statement Show that there are continuous functions g:[-1,1]\to R such that no sequence of polynomials Q_n satisfies Q_n(x^2)\to g(x) uniformly on [-1,1] as n\to\infty The Attempt at a Solution Suppose there is a sequence Q_n such that Q_n(x^2)\to g(x) uniformly for g(x)=x. Then...- holomorphic
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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Equicontinuity and uniform convergence
Homework Statement This is a nice one, if it's correct. Show that if (fn) is a sequence of elements of C(X, Y) (where Y is a metric space) which converges uniformly, then the collection {fn} is equicontinuous. The Attempt at a Solution Let ε > 0 be given and let x0 be a point of X...- radou
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
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B
Uniform convergence of integrals
Homework Statement Hi All, I've been having great difficulty making progress on this problem. Suppose gn converges to g a.e. on [0,1]. And, for all n, gn and h are integrable over [0,1]. And |gn|\leqh for all n. Define Gn(x)=\intgn(x) from 0 to x. Define G(x)=\intg(x) from 0 to x...- bbkrsen585
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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M
Uniform convergence of a series.
Homework Statement Prove the series converges uniformly on R to functions which are continuous on R. \sumn\geq0 (-x)2n+1/(2n+1)! Homework Equations The Attempt at a Solution I'm having trouble actually figuring out what to use for this series.. It looks like a Taylor series...- missavvy
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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S
Uniform Convergence of Continuous Functions: A Proof?
Homework Statement As in the question - Suppose that f_n:[0,1] -> Reals is a sequence of continuous functions tending pointwise to 0. Must there be an interval on which f_n -> 0 uniformly? I have considered using the Weierstrass approximation theorem here, which states that we can find...- simba31415
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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Uniform convergence of piecewise continuous functions
I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous? I've thought of... -
V
Pointwise and uniform convergence of fn(x) = (1-x)^(1/n)
Homework Statement Let fn(x) = (1-x)^(1/n) be defined for x an element of [0,1). Does the sequence {fn} converge pointwise? Does it converge uniformly? Homework Equations The formal definition of pointwise convergence: Let D be a subset of R and let {fn} be a sequence of real valued...- Vespero
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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E
Show Uniform Convergence of fn to f in Compact Metric Space
Homework Statement Alright, here is the problem. Given a compact metric space X, and a sequence of functions fn which are continuous and f_{n}:X->R (reals), also f_n->f (where f is an arbitrary function f:X->R). Also, given any convergent sequence in X x_{n}->x, f_{n}(x_{n})->f(x). The problem...- Enjoicube
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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N
Does This Series Converge Uniformly on [0, ∞)?
Homework Statement Does the following series converge uniformly? [sum from n=1 to inf] \frac{e^{-nx}}{n^2} on [0, inf) Homework Equations I know I need to use the M test or Cauchys Principle of uniform convergence. My tutor suggests using the former if there is uniform convergence...- Namo
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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F
Pointwise vs. uniform convergence
A function is pointwise bounded on a set E if for every x\in E there is a finite-valued function \phi such that |f_n(x)|<\phi(x) for n=1,2,.... A function is uniformly bounded on E if there is a number M such that |f_n(x)|<M for all x\in E, n=1,2,.... I understand that in uniform...- ForMyThunder
- Thread
- Replies: 4
- Forum: Calculus
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W
Uniform convergence of a series
Homework Statement We know that f is uniformly continuous. For each n in N, we define fn(x)=f(x+1/n) (for all x in R). Show that fn converges uniformly to f. Homework Equations http://en.wikipedia.org/wiki/Uniform_convergenceThe Attempt at a Solution I know that as n approaches infinity...- WaterPoloGoat
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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Uniform convergence - Series of functions
Homework Statement \mbox{Check whether } \sum_{n=0}^\infty \frac {1}{e^{|x-n|}} \mbox{ is uniform convergent where its normaly convergent} The Attempt at a Solution \mbox{I choose } \epsilon = 1/2 a_n=\frac {1}{e^{|x-n|}}\ ,\ b_n= \frac {1}{n^2} \lim_{n\rightarrow\infty} \frac {a_n}{b_n}=0\...- estro
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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Proving Uniform Convergence of f_n(x) in [a,b]
Homework Statement Suppose: f(x)\ and\ f'(x)\ are\ continuous\ for\ all\ x \in R For\ all\ x \in R\ and\ for\ all\ n \in N\ f_n(x)=n[f(x+\frac{1}{n})-f(x)] Prove\ that\ when\ a,b\ are\ arbitrary,\ f_n(x)\ is\ uniform\ convergent\ in\ [a,b] The Attempt at a Solution\lim_{n\rightarrow...- estro
- Thread
- Replies: 32
- Forum: Calculus and Beyond Homework Help
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M
Can all subsequences of a function sequence in C([0,1], R) converge uniformly?
Let y_n be a sequence of functions in \mathcal{C}([0,1], \mathbb{R}) Suppose that every subsequence of y_n has a subsequence that converges uniformly. Prove that they all converge to the same limit. -
R
Uniform convergence and continuity
1.kn (x) = 0 for x ≤ n x − n, x ≥ n, Is kn(x) uniformly convergent on R? I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it? 2.Fix 0 < η < 1. Suppose now...- rainwyz0706
- Thread
- Replies: 5
- Forum: Calculus and Beyond Homework Help
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L
How Can Uniform Convergence Be Used to Approximate Continuous Functions?
let f is a continuous, real-valued function on [a,b] then, for any e, there exist a polygonal function p such that sup|f(x)-p(x)|<e using uniform convergence, this might be shown... but i cannot figure it out... -
C
Using Dirichlet's test for uniform convergence
Homework Statement Show that \sum(-1)^(n+1) / (n + x^2) converges uniformly but not absolutely on R. Homework Equations Using Dirichlet's Test for uniform convergence. (fn) and (gn) are sequences of functions on D satisfying: \sum fn has uniformly bounded partial sums gn -> 0...- cchatham
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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B
Pointwise Limits and Uniform Convergence Help
Hi all Came across this question which i have attempted to answer but am sure i have gone wrong somewhere, any help would be appreciated... Suppose fn(x):R-->R, where fn(x)=x+(1/n) (n belongs to N). Find the pointwise limits of the sequences (fn), (1/n fn) and (fn2). In each case determine...- BlackTiger
- Thread
- Replies: 1
- Forum: Calculus
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C
Uniform convergence of function sequence
Homework Statement For k = 1,2,\ldots define f_k : [0,1] \to \mathbb{R} by \begin{align*} f_k(x) = \left\{ \begin{array}{ll} 4k^2x & 0 \leq \displaystyle x \leq \frac{1}{2k} \\ 4k(1 - kx) & \displaystyle \frac{1}{2k} < x \leq \frac{1}{k} \\ 0 & \displaystyle \frac{1}{k} < x \leq 1...- complexnumber
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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K
Uniform Convergence & Boundedness
"Let fk be functions defined on Rn converging uniformly to a function f. IF each fk is bounded, say by Ak, THEN f is bounded." fk converges to f uniformly =>||fk - f||∞ ->0 as k->∞ Also, we know|fk(x)|≤ Ak for all k, for all x But why does this imply that f is bounded? I don't see why it...- kingwinner
- Thread
- Replies: 7
- Forum: Calculus
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K
Series of functions & uniform convergence
Homework Statement Homework Equations The Attempt at a Solution This is not graded homework, but optional exercises I found in my textbook. It's days before my exam, but I'm still not sure how to do problems like this. I would really appreciate if someone can teach me how to solve...- kingwinner
- Thread
- Replies: 18
- Forum: Calculus and Beyond Homework Help
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C
Uniform convergence and continuity
Homework Statement Theorem: Let (X,d_X),(Y,d_Y) be metric spaces and let f_k : X \to Y, f : X \to Y be functions such that 1. f_k is continuous at fixed x_0 \in X for all k \in \mathbb{N} 2. f_k \to f uniformly then f is continuous at x_0. Homework Equations If all f_k are...- complexnumber
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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C
Pointwise convergence and uniform convergence
Homework Statement Define the sequence \displaystyle f_n : [0,\infty) \to \left[0,\frac{\pi}{2}\right) by f_n(x) := \tan^{-1}(nx), x \geq 0. Homework Equations Prove that f_n converges pointwise, but not uniformly on [0,\infty). Prove that f_n converges uniformly on [t, \infty)...- complexnumber
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
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N
Using the Weierstrass M-test to check for uniform convergence
Homework Statement \text{Show that the series }f(x) = \sum_{n=0}^\infty{\frac{nx}{1+n^4 x^2}}\text{ converges uniformly on }[1,\infty]. Homework Equations The Attempt at a Solution I think we should use the Weierstrass M-test, but I'm not sure if I'm applying it correctly: For...- nietzsche
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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S
Uniform Convergence, help with proof of theorems
Homework Statement 1.) Prove that if { f_{n} } is a sequence of functions defined on a set D, and if there is a sequence of numbers b_{n}, such that b_{n} \rightarrow 0, and | f_{n}(x) | \leq b_{n} for all x \in D, then { f_{n} } converges uniformly to 0 on D. 2.) Prove that if { f_{n} } is a...- spenghali
- Thread
- Replies: 4
- Forum: Calculus and Beyond Homework Help
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K
Another uniform convergence question
Homework Statement h:[0,1] -> R is continuous Prove that t(x) = \sum^{infinity}_{n=0} xnh(xn) is uniformly convergent on [0,s] where 0<s<1 Homework Equations The Attempt at a Solution I have the definition of h being continuous but after this I am pretty clueless about how to...- Kate2010
- Thread
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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K
Uniform Convergence Homework: Is fn(x) Convergent on [a,b] & R?
Homework Statement fn(x) = 0 if x \leq n = x-n if x\geq n Is fn(x) uniformly convergent on [a,b]? Is it uniformly convergent on R? Homework Equations The Attempt at a Solution I think limn -> infinity fn(x)= 0 However I do not know what supx in [a,b] |fn(x)| would be, I...- Kate2010
- Thread
- Replies: 10
- Forum: Calculus and Beyond Homework Help
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R
Inverting Consequences of Uniform Convergence
Hi. Now you probably know that if a function fk(x) converges uniformly to f(x) then we are allowed to certain actions such as limn-> \inf \int f (of k) dx = \int f dx In other words we are allowed to exchange limit and integral. Now say we have any sequnce valued function fk(x) . And we... -
F
Uniform Convergence Problem: Does fn converge uniformly to f?
I'm trying to better understand convergence so I made upa problem for myself based on an example from class. I want to know if I'm answering my own questions correctly. Define a sequence of functions fn(x) = 1 if x is in {r1, r2, ... , rn} and 0 otherwise. Where r1, r2, ... , rn are the first...- futurebird
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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Uniform Convergence: Does Not Converge on (0,1)
Homework Statement Let fn(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly. Homework Equations The Attempt at a Solution I know why it is not uniformly convergent. Even though fn(x) goes to zero monotonically on the interval (0,1), it's not...- hth
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- Forum: Calculus and Beyond Homework Help