# Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

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### Question on dominated convergence theorem.

I don't understand a statement in a paper I'm reading: We have a function $f:\mathbb{R}^n \rightarrow\mathbb{R}$. $f$ is convex and $\exp(-f)$ is a probability density function but I don't think that ...
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### Prove that $\lim_{n\to\infty}Q_n(x)=g(x)$ for some polynomial $Q_n$.

Suppose that for any complex function it continues on $[0,1]$ with $f (0) = f (1) = 0$. there exists a sequence of polynomials $P_n$ such that $\lim_{n \to \infty} P_n (x) = f (x)$ uniformly ...
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### Integral of $L^{p}(\mathbb{R}^{n})$ sequence outside a large ball

I've seen the question Integral of L^1 function outside a large ball Could it be generalized for a sequence? Specifficaly, let $(u_{j})$ a bounded sequence in $L^{p}(\mathbb{R}^{n})$, I would like to ...
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### Given $x_{n+1}=x_n(1-x_n)$, find $\lim_{n \to \infty}n.x_{n}$ [duplicate]

Let the sequence $\{x_n\}$ be such that $0<x_{1}<1$ and $x_{n+1}=x_{n}\left(1-x_{n}\right)$. Find $\lim_{n \rightarrow \infty}n.x_{n}$ So far I have only been able to prove that the sequence is ...
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### If $f:B\to\mathbb{R},\ B\subset\mathbb{R}$ is increasing then there is a sequence of strictly increasing functions whose pointwise limit is $f$

I have proved the following statement and I would like to know if my proof is correct and/or how it could be improved. Suppose $B\subset\mathbb{R}$ and $f:B\to\mathbb{R}$ is an increasing function. ...
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### Show uniform convergence of sequence of functions $f_n(x)=x^n$ on $[0,0.9]$

Show uniform convergence of sequence of functions $f_n(x)=x^n$ on [0, 0.9]. There is a similar question for the interval [0, 1) (https://math.stackexchange.com/questions/2191621/is-the-sequence-of-...
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### Proving $f_n(x)$ not uniformly convergent

Let $f_n(x)$ be a sequence of functions that converge for $f(x)$ on $x\in[a,b]$ but it's not uniformly convergent on the same range of $x$. Prove that $f(x)$ is not uniformly convergent on $x\in(a,b)$....
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### $f_n\rightarrow f$ uniformly in an interval

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ a sequence of functions and $f:\mathbb{R}\rightarrow \mathbb{R}$. Which of the following statements are correct? (a) If $f_n\rightarrow f$ uniformly in each ...
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### Show that sequence of antiderivatives has a subsequence that converges pointwise

I have this question from a real analysis assignment, For $n\geq1$, let $f_n:[0,1]\to\mathbb{R}$ be a continuous function with $$|f_n(x)|\leq 1+\frac{n}{1+n^2x^2}$$ Define $F_n:[0,1]\to\mathbb{R}$ via ...
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### Prove $\sum_{n=1}^\infty(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}})$ convergent

Let $(a_n)_{n=1}^\infty$ Let be a positive, increasing, and unbounded sequence. Prove that the series: $$\sum_{n=1}^\infty\left(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}\right)$$ convergent. We know that ...
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### uniform convergence of $f_n(x)=1/n^x$

Consider the sequence of functions: \begin{align} f_n(x) = \frac{1}{n^x}. \end{align} Does $(f_n)_n$ converge uniformaly on $[0,\infty)$. I proved via the Cauchy criterion that $(f_n)_n$ can not ...
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### differentiable and sequences of functions [closed]

Let $f_n:(a,b)\rightarrow \mathbb{R}$ be differentiable and $f_n\rightarrow f$. Can $f$ be differentiated? Would it be $f'(x_0)=\lim_{n \to \infty}f_n'(x_0)$ for $x_0\in (a,b)$ ?
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### $f_n(x)= f(x+n)$ show that the limit function is uniformly continuous

Let $f$ be a real-valued continuous function on $I=\{x\in \mathbb{R} | x \geq 0\}$. For a positive integer $n$ the function on $I$ is defined by \begin{align*}f_n(x)= f(x+n)\end{align*} Answer the ...
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### pointwise convergence does not imply uniform convergence for series

Regarding sequences of functions $(f_n(x))$, I can wrap my head around the idea that uniform convergence $\Rightarrow$ pointwise convergence, but pointwise convergence does not imply uniform ...
I need to find if the series $$\sum_{k=2}^\infty\left(\cos\frac{x}{k}-\cos\frac{x}{k-1}\right)$$ converges uniformly on $(-\infty,\infty)$. My attempt: The partial sum of the series: \sum_{k=2}^n\...