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.

A sequence of functions is denoted as $\{ f_n \}$ or $(f_n)$. For example,

  1. $f_n:\Bbb R \to \Bbb R$ defined by $f_n(x)=\frac {1}{1+nx^2}$
  2. $f_n:\Bbb R \to \Bbb R$ defined by $f_n(x)=e^{-nx^2}$
  3. $f_n:\Bbb C \to \Bbb C$ defined by $f_n(z)=\frac {\sin nz}{\sqrt n}$

are sequences of functions. Usually, our first encounter with them is to study their nature of convergence.

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Example of sequence of functions that are differentiable and converge uniformly to |x| on [-1,1]

Is there an example of sequence of functions $f_n$ that are differentiable and converge uniformly to $|x|$ on $[-1,1]$?
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A sequence of continuous functions

There exist a sequence of continuous functions $\{f_n\}$ on $\mathbb{R}$ such that $\{f_n\}$ converges to $f$ uniformly on $\mathbb{R}$, but $$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}f_n(x)dx \neq \int_{-\infty}^{\infty}f(x)dx$$ solution I…
TheStudent
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Uniform and absolutely convergence of functions

Is there any relationship between absolutely convergence and uniform convergence of function . I know a sequence of function which converge uniformly but not absolutely . Please provide example or proof of converse part.
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