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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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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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Turning a continuous everywhere differentiable nowhere function into a smooth function by infinitely many times definite integration?

Let $W(x)$ be a real-vlued function defined on a (possibly infinite) interval $\text{T}\subseteq\mathbb{R}$ containing $0$ that is continuous everywhere differentiable nowhere on $\text{T}$. Define ...
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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 ...
312 views

$\epsilon$/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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Is there a short form of a polynomial function applied to itself $i$-times?

Given a polynomial $f$ of degree $m$: $$f(x)=\sum_{j=0}^{m} a_jx^j$$ Now this polynomial is applied to itself $f(f(f(f...(f(x))))$ for $i$($-1$) times $->f^i(x)$. The resulting function is a ...
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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 ...