# 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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### Is there any $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ such that $\frac{d}{dt}\|K_t^{\epsilon}\|^2\to 1$?

I am looking for conditions or at least one example such that the following holds true. Let $K_t^{\epsilon}:[0,T]\to\mathbb R$ be an smooth function such that $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ in ...
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### Transitivity of convergence of sequence in $L^1$ and $BV$ [closed]

Let $(f_n)_n\in BV([0,T])$ a sequence of functions of bounded variation converging to some $f\in BV([0,T])$, that is to say $$\lim_{n\rightarrow+\infty}\parallel f_n-f\parallel_{BV([0,T])} = 0.$$ and ...
1 vote
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### Uniform convergence of sequence of functions that converges pointwise to an unbounded function

I have the following sequence of functions before me: $f_n(x)=\dfrac{1-x^n}{1+x},-1<x<1$ I have to determine whether above sequence of functions is uniformly convergent or not. First of all, I ...
1 vote
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### Uniform convergence of a sequence of functions on $[0,1]$ as follows [duplicate]

Let $f_n:[0,1]\to \Bbb R$ be a sequence of functions defined by \begin{align*} f_n(x):= \begin{cases} nx, 0\le x \le \frac1n \\ 1, \frac1n \le x \le 1 \end{cases}. \end{align*} Show that $(f_n)$ is ...
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### Show that the sequence of functions $(f_n)$ is bounded uniformly. [duplicate]

I am trying to show that the sequence $$f_n(x)=\sum_{m=1}^n sin(mx)$$ is uniformly bounded on $[a,2\pi-a]$ with $0<a\le\pi$. From playing around with the graph of $f$ its pretty clear that the ...
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1 vote
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### Uniform convergence and rigor proof of pointwise convergence of $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$ for all $x \in (0,\infty)$.

Let $(f_n)$ be a sequence of functions where $f_n:(0,\infty) \to \Bbb R$ defined by $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$. Show that $f_n$ does not converge uniformly on $(0,\infty)$. My ...
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### Sequence of functions $f_n(x) = \frac{\sin nx}{1+nx}$ converges to $0$.

Let $(f_n)$ be a sequence of functions with $f_n : [a,\infty) \to \Bbb R$ defined by $f_n(x) = \frac{\sin nx}{1+nx}$, for all $x \in [a,\infty)$ and $n \in \Bbb N$. Show that $f_n$ converges pointwise ...
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1 vote
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### Find $\sum f_n(x)$

Problem For non-negative integer $n$, define $$f_n\colon [0, 1]\to\mathbb{R}, \quad f_n(x)=\int_0^x f_{n-1}(t)dt\quad (n>0),\quad f_0(x) = e^x$$ Find $g(x)=\displaystyle\sum_{n=0}^\infty f_n(x)$ ...
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1 vote
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### Showing that the given sequence of functions converges to a given function on the given set as follows

Let $(f_n): \Bbb R \to \Bbb R$ be a sequence of functions defined by $f_n(x) := x^n$ for $x \in \Bbb R$ and $n \in \Bbb N$. Let $f:(-1,1] \to \Bbb R$ be a function defined by \begin{align*} f(x):= \...
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### Convergence of Sequence of functions with nested radicals.

I am reading Lascota and Mackey's "Chaos, Fractals, and Noise" (which is great). In an early chapter, they define the following operator (really this is a special case of the Frobenius-...
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### Calculate $\lim_{n\to\infty} \frac{n\sin(x/n)}{x(x^2+1)}$ in $\mathbb{R}^+$

For all $n\in\mathbb{N}$, let $f_n:(0,\infty)\to\mathbb{R}$ be the function defined by $f_n(x)=\frac{n\sin(x/n)}{x(x^2+1)}$. Find the pointwise limit of $(f_n)$. I know that for all $x>0$ we have ...
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### Showing that this particular sequence of functions defined by recursion is point-wise bounded

Let $(s_n)_n$ be a summable sequence. Consider the following sequence of functions $(z_n)_{n\geq 0}$, $z_n:[0,+\infty)\rightarrow\mathbb{R}$, defined recursively as \begin{align*} z_0(x)&\equiv 0\\...
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### Riemann integral of product of sequence of function with a continuous function at a point.

Here is the setup to a questions I've been looking at. I mainly just need a hint on how to proceed from where I'm at. Or suggest a theorem/corollaries that would help. The problem: Let $\{g_n\}$ be a ...
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### Existence of closed form formulas for integer sequences

Suppose that z x y ———————— 0 1 0 1 2 0 2 2 1 3 3 0 4 3 1 5 3 2 6 4 0 7 4 1 8 4 2 9 4 3 . . . . . . . . . n xn yn the z column ...
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### Prove polynomic sequence with each $P_i$ containing one critical point and all intersecting at know $x$ define a convergent sequence of arc lengths
Have a sequence of polynomials, the initial $p_0$ is a quadratic, with degree of $p_i$ greater than $p_{i-1}$. Sturm analysis of each $p$'s derivative suggests one critical point. They are defined to ...
Let $(a,b)$ be an open interval in $\mathbb{R}$. $f:(a,b)\rightarrow\mathbb{R}$ $z\in (a,b)$ and let $f$ be continuous in $z$. Let $f$ be differentiable over $(a,b)\setminus\{z\}$. My question is ...