# 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.

61 questions
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### Convergence in measure iff every subsequence has a convergant subsequence

My task is to show that on a finite measure space $(X,A)$, if $f_n \to f$ in measure, then every subsequence of $\{f_n\}$ has a subsequence that converges to $f$ almost everywhere. I believe I was ...
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### Sum of an odd recursive sequence

Let $a_0 = 1$ $a_1 = 1 - \frac{e}{2}$ $a_n = \frac{e}{2^n} - \frac{1 - a_{n-1}}{n - 1}$ for $n > 1$. Find $\sum_{r=0}^{\infty}a_r$.
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### Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
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### Finding sequence of functions with compact support for Integral of given function

Let $a>0$ and $f(x):\mathbb{R}\rightarrow\mathbb{R}$ with $$f(x):\begin{cases}\frac{1}{\sqrt{a^2-x^2}},& x\in(-a,a)\\ 0,&\text{else} \end{cases}$$ I now have to construct sequences of ...
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### Can i prove the limit of a function using the definition of sequence limits?

So I'm asked to prove the $\lim_{x\to 1} \frac{4x+3}{x^2+4x-3}$ and of course it can be easily found $\frac{7}{2}$. However, I know I can prove it with epsilon delta definition but frankly that's ...
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### Show that $f_n (x)$ converges to $f(x)=0 \ \forall x$

$f_n$ is defined on $[0,1] \ \ \forall n \in \mathbb{N}$, $f_n(x)=\begin{cases} n(1-nx) &\mbox{if } \ 0 \leq x < \frac{1}{n} \\ 0 & \mbox{if } \ \frac{1}{n} \leq x \leq 1 \end{cases}$ ...
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### Example of a sequence $(f_n)_n$, such that $(f_ n')_n$ converges uniformly, but $(f_n)_n$ doesn't converge in any point.

I've seen the following theorem: If $(f_n)_n$ is a sequence of $C^1$ functions in $[a,b]$, such that $(f_n(c))_n$ converges to $f(c)$ for some $c\in [a,b]$, and $(f_n')_n$ converges uniformly to $g$...
I have a function $g(\cdot)$ such that such that $\lim\limits_{t \to \infty}g(t-s)=0$. I need to show that $\lim\limits_{T \to \infty}\int_{-\infty}^tg(T-s)ds=0$, where $0<t\ll T$. I thought that ...
Let $f_n(x)=\displaystyle \frac {x^n}{1+x^2}$, for $n\in \Bbb N$. Then which of the following statements is true? (A)$\;\;f_n$ converges uniformly on $[0,1]$. (B)$\;\;f_n$ converges uniformly on \$[-...