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.

248 questions
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Finding the limit of a sequence of integrals

Let us define a sequence of function as $$f_n(x)=\frac{2nx^{n-1}}{x+1}\;\;\text{for each x\in [0,1] and for all n\in\mathbb{N}}$$ What is $\displaystyle \lim_{n\to \infty} \int_0^1 f_ n(x) dx$ ...
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Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
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Discontinuity properties of $f_n$ carries over to the limit function $f$

Suppose that $f_n:[a,b] \rightarrow \Bbb R$ and $f_n$ converges uniformly to $f$. Which of the following discontinuity properties of the functions $f_n$ carries over to the limit function ? ...
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Is it true that $\lim_{n\to\infty}\int_{0}^{1}f_n(x)\,dx = \int_{0}^{1}f(x)\,dx$ in general and if $|f_n(x)|\le 2017$?

Let $f_n(x)$ and $f(x)$ be continuous functions on $[0, 1]$ such that $\lim_{n\to\infty} f_n(x) = f(x)$ for all $x \in [0, 1]$. Answer each of the following questions. If your answer is “yes”, then ...
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Uniform convergence on two separate intervals

Test the sequence of functions for uniform convergence $$f_n(x)=\frac{nx}{1+n^2x^2}$$ $a) \text{on} \ [0,1]$ $b) \text{on} \ [1,{\infty})$ a) If we differentiate this with respect to ...
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$\epsilon$/3 - Proof for converging sequence of functions

I tried to prove the following statement and would like to ask if it's okay like that. $\underline{Statement}$: Let [B,$||.||_B$] be a complete normed linear space, $T_n :B \rightarrow B$ a ...
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Arzelá-Ascoli Theorem precompact sets

Is the Arzelá-Ascoli Theorem true in a precompact subset of $\mathbb{R}^n$?. If $S \subset \mathbb{R}^n$ is precompact and we have a sequence $(f_n)$ of functions in $C(S)$ (Space of bounded and ...
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Calculate the limit of $\frac{n^{n+1}}{n!}\int_0^xe^{-nt}t^n\,\mathrm dt$

Set $R_n(x)$ the quantity in the title, for $n\in\mathbb{N}$ and $x > 0$. I'm trying to prove that if $x < 1$, then $R_n(x)\underset{n\rightarrow +\infty}{\longrightarrow} 0$, if $x > 1$, ...
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Elementary proof regarding $\sum_{n=1}^\infty \frac{\sin(nx)}{n}$

I was looking for an elementary proof (without use of Fourier series) that $$\sum_{n=1}^\infty \frac{\sin(nx)}{n}$$ converges to $(\pi-x)/2$ for $x\in (0,2\pi)$. I have managed to demonstrate that ...
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A fixed point theorem for continuous and increasing function on $[0,1]$

Let $f:\ [0,1] \to [0,1]$ be continuous and increasing. Also let $p \in [0,1]$. Define $$x_n = f^n(p) = f(f(...f(p))...)$$ Prove that either $p$ is a fixed point of $f$ or $x_n$ converges to a ...
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Limit of integrals of trigonometric functions

If we have a sequence of functions defined by $f_n(x) = \frac{\sin((x + \frac 1n)^2) - \sin((x-\frac 1n)^2}{\sin(\frac 1n)},$ how can we find $\lim_{n \rightarrow \infty} \int_0^1 f_n(x) \, d\lambda$ ...
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Sequence of differentiable,equicontinuous functions

I got stuck the other day trying to tackle the following problem : Let $\left \{ f_n\left. \right \} \right.$ be a sequence of differentiable functions : $f_n \quad : [0,1] \to \mathbb{R}$ ...
Let $(f_{n})_{n}$ be a sequence of functions $f_{n}: \mathbb{R} \to \mathbb{R}$. They converge pointwise to a function $f: \mathbb{R} \to \mathbb{R}$. We know that the convergence is uniform on ...