# 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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### 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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### Showing that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not converge uniformly

I am trying to show that $f_n(x) = \frac{nx}{1+n^2x^2}$ where $x\in \mathbb{R}$ does not converge uniformly. I have made an attempt and would like to make sure that I am going about it in the right ...
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### Constructing a sequence of function [closed]

Construct a sequence of functions on [0,1] each of which is discontinuous at every point on [0,1] and which converges uniformly to a function that is continuous at every point?
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### Limiting total variation attached to sequence of uniformly vanishing functions of bounded variation

Let $(f_n)_n$ be a sequence of real functions of a single real variable with compact support in $[0,1]$ and of bounded variation all of them. Let the sequence be uniformly convergent to $0$. Is it ...
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### continuity of the function $f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}$

I need to check the continuity and differentiability of the function $f(x)$ at $x=0$ where, $$f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}.$$ I tried to check the domain of ...
### Show that the sum of reciprocal products equals $n$
I don't even know how to proceed. Please help me with this. (Original at https://i.stack.imgur.com/DRIX8.jpg) Consider all non-empty subsets of the set $\{1, 2, \ldots, n\}$. For every such ...
Let $\{f_{n}\}$ be a sequence of functions differentiable at $x_0$. Let $r_{n}(h,x_0) = f_n(x_0+h) - f_n(x_0) - f'_n(x_0)h$ We know that for any fixed $n$, $r_n(h,x_0) = o(|h|)$ as $h \to 0$ from ...