# 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 functions that fulfil a condition and are strictly increasing.

I need to find $\lambda_{n}(t)$ such that it gives $0$ for $t = 0$ and $p + \dfrac{1}{n}$ for $t = p$ and $1$ for $t = 1$. The first case is fulfilled but I can't seem to find a function that would ...
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### Show that $\sum_{1}^\infty\frac{\sin(nx)}{n^3}$ is differentiable everywhere

I have recently been trying out some questions on series of functions.In one of the questions, I was given a series $$\sum_{1}^\infty\frac{\sin(nx)}{n^3}$$ and now I am supposed to show that the above ...
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### Uniform Convergence of a series of functions using the Dirichlet's test

I have recently been trying out some questions on series of functions. I got stuck in one of those problems in which I am supposed to show that the below series of functions is uniformly convergent on ...
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### Limit of a function for $\epsilon>1$.

I have been trying some questions on the convergence of a sequence of functions and was wondering about an intermediate step in which we have $\epsilon\gt0$ and an $x$ such that $0<x<1$ and it ...
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### Uniform convergence of $x^n$ using the definition

I have been trying to prove the uniform convergence of sequence of functions defined by $f_n(x)=x^n$ on $[0,k]$ where $k<1$ by the epsilon definition of uniform convergence. I have found the point-...
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### Uniform Convergence of $\frac{n}{x+n}$

I was trying an exercise on uniform convergence of sequence of real-valued functions. I got stuck in a problem in which I am supposed to prove that sequence defined by $f_n(x)=\frac{(n)}{(x+n)}$ is ...
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### uniform convergence of functional series $\sum_{k=1}^{\infty} {(-1)^{k} \frac{k+\sin(x)}{k^{2}}}$

I tried to solve the exercise that ask to define the convergence and the uniform convergence of the functional series $$\sum_{k=1}^{\infty} {(-1)^{k} \frac{k+\sin(x)}{k^{2}}}$$ it is easy to prove ...
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### Find a limit for a sequence of functions with the domain in $(0, \infty)$

How to find limits for function $f_n = \sqrt{n}\left(\sqrt{x - \frac{1}{n}}- \sqrt{x}\right)$ if $Df \in (0,\infty)$ I think as $n \rightarrow \infty$ it would be $\infty$ and then no matter the $x$ ...
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### Continuity of the sequence of averages

Suppose you have a sequence of bounded continuous functions $f_n:(a,b]\to\mathbb{R}$. Then, define the function $$S(x)=\liminf_{n\to\infty}\frac{1}{n}\sum_{j=1}^nf_n(x).$$ Is $S(x)$ continuous?