All Questions
Tagged with sequences-and-series uniform-convergence
1,362
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exchange order of limit of random variable
Suppose to have a sequence of discrete random variables $X_n(\lambda)$ depending on some parameter $\lambda \ \in [0,\infty]$ and $n \in \mathbb{N}$ and that the following limits hold almost surely:
$...
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Prove function defined by series is continuous [duplicate]
I am working on this problem. I am studying for an exam. I find it hard to believe this has not been asked before, but I could not find it.
Show that the function $f(x) = \sum_{n=1}^{\infty} \frac{1}{...
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Two series converges pointwise to same limit, and one converges uniformly
We have $\sum{f_n}$ and $\sum{g_n}$.
$f_n,g_n$:[0,1]->$\mathbb{R}$ and all f are non-negative and all g are continuous. Both series converges pointwise to the same limit. How to prove that if $\sum ...
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Uniform Convergence of a Sequence of Differentiable Functions
I'm currently studying real analysis and I've come across a problem that I'm having trouble with. The problem is as follows:
Let $(\phi_n)$ be a sequence of differentiable functions on $[a,b]$ such ...
user1055322
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Uniform Convergence of a Sequence of Functions and Differentiability
I am currently studying the topic of uniform convergence and its implications on differentiability. I came across a problem that I have been trying to solve but I am stuck. The problem is as follows:
...
user1319676
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43
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Convergence and Differentiability of a Sequence of Functions
Question:
I'm studying a sequence of functions $$f_n(x) = \sqrt{x^2 + \frac{1}{n}}$$ defined on the domain $[-1,1]$ and I'm trying to understand their behavior as $n$ approaches infinity.
Context and ...
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Understanding Uniform Convergence of a Sequence of Functions
I am currently self-studying and came across the following theorem:
Suppose that a sequence of functions $\phi_n$ converges uniformly to $0$ on $[a,b]$. Now suppose we have a sequence of functions $...
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Prove a series is uniformly convergent, hence continuous. [duplicate]
Question: Show that the series $\sum_{n=1}^{\infty}(ne^{-nx})$ is continuous on $(0,\infty)$.
What I have tried: Since $ne^{-nx}$ is continuous for all $x>0$, it suffices to prove the uniform ...
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Let $f_n:[0,1] \to \mathbb {R}$ with $f^{'}_n$ uniformly bounded by $1$ and exists $f$ s.t $f_n \to f$ pointwise. Prove convergence is uniform.
Question:
Let $f_n:[0,1] \to \mathbb {R}$ with $f^{'}_n$ uniformly bounded by $1$. Let $f:[0,1] \to \mathbb R$ s.t $f_n \to f$ pointwise in $[0,1]$. Prove convergence is uniform.
My attempt:
So for ...
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Proof of $K_0(z)=-\left(\log\frac{z}{2}+\gamma\right)I_0(z)+\sum_{n=1}^\infty \frac{(z/2)^{2n}}{n!^2}H_n$
Let $I_{\nu}$ be the Bessel I function of order $\nu$ defined by
$$I_{\nu}(z)=\sum_{n=0}^\infty \frac{(z/2)^{2n+\nu}}{n!\Gamma (n+\nu+1)}$$
and let $K_{0}$ be the Bessel K function of order $0$ ...
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Showing a Fourier series is continuously differentiable
Let $f:\mathbb R\to\mathbb R$ be a $2\pi$-periodic continuously differentiable function. Define Fourier coefficients:
$$c_n=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}dx$$
$$c_n'=\frac{1}{2\pi}\int_{-\...
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Understanding a proof that a uniformly Cauchy sequence of continuous functions $\{f_n\}_n$ converges uniformly to a limit function $f$
Suppose $\{f_n\}_n$ is a uniformly Cauchy sequence of continuous function $f_n:\mathbb{R}\to S$ for a complete metric space $S$. I am trying to understand the "standard" proof that the ...
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Proving that the series $\sum_{n=1}^{\infty}f_n(1)$ converges [closed]
I am working on an exercise that goes like this: consider the functions $f_n:[0,1]\rightarrow\mathbb{R}$ for $n\in\mathbb{N}$ such that they are continuous and that $\sum_{n=1}^{\infty}f_n(x)$ ...
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How to construct a complex function series that converges uniformly on a closed region but its derivative does not converge uniformly on the boundary?
I'm looking for an example of a series of complex functions $\{f_n(z)\}$ with the following properties:
The series $\sum_{n=1}^{\infty} f_n(z)$ converges uniformly on a closed region $D$ in the ...
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Alternative proof for Dini's theorem. Is it correct?
Let $(X,d)$ be a compact metric space, $f_n : X \to \mathbb{R}$ a sequence of continuous functions, $f_{\infty} : X \to \mathbb{R}$ a continuous function. Now suppose that $f_n \to f$ pointwise, and ...
