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.

249 questions
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Pointwise convergence of $f_n(x) = \left(\frac{3^{x/n} + e^{x/n}}{2} \right)^{-n}$

I am given the sequence $f_n : [0,\infty) \to \mathbb R$ such that $$f_n(x) = \left(\frac{3^{x/n} + e^{x/n}}{2} \right)^{-n}$$ and I am asked to discuss its convergence almost everywhere w.r.t. the ...
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Convergence of Sequences of Functions Evaluated at Sequences in Their Domain

Suppose (1) A sequence of continuous functions$f_n(x)$ converges to a continuous function $f(x)$ pointwise on some set $I$, and (2) a sequence $\{x_n\}$ converges to $x \in I$ (****) Is it true ...
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Differentiable limit of a (uniformly convergent) sequence of differentiable function (again, but not exactly !)

If one has a uniformly convergent sequence of differentiable functions $f_n$ (say from $\mathbb{R}$ to $\mathbb{R}$), we know that the limit $f$ is not always differentiable. Even if it is ...
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Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. [duplicate]

Consider the function $f$ which is continuous. Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. Here first I attempted to prove $f_n=nx^n$ is uniformly convergent using sup-norm limit but ...
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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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Is it a continuous?

For each $n\in\mathbb{N}$, define a function $f_n:[0,1]\to\mathbb{R}$ by $$f_n(x)=\int_{1/n}^{1}\frac{t^{x}}{\sqrt{t+x}}\,dt.$$ Then, is it continuous for each $n\in\mathbb{N}$? I don’t understand ...
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Uniform convergence of a sequence of functions 4

Prove that the sequence $\left((nx)/(1+4n^2x^2)\right)_{n\in\mathbb N}$ is not uniformly convergent on $(-a,a)$, where $a > 0$ My attempt: $\lim_{n\to\infty}(nx)/(1+4n^2x^2) = 0 = f(x)$ Now, ...
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Pointwise convergence of $h_{n}(x)$ on [0,$\infty$)

I know that it converges pointwise to $1$ if $x>0$ and to $0$ if $x=0$ using limits . But I am struggling to show this formally. Any help would be greatly appreciated . Thanks
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How to show a function is locally C^1 implies globally C1?

Actually there is a series problem like f(x)=sum(n=1 to ∞)[sin(nx^2)/1+n^3], the question was whether f(x) is C^1 or not. This question has already answered, but a big issue of mine is I can't find ...
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Determine the sequence generated by the following exponential generating functions:

a) $f(x)=3e^{3x}$ I have \begin{align}f(0)&= 3 \\ f(1)&=3e^3 \\ f(2)&=3e^6 \end{align} So would my sequence be $a_n=3e^{2n}$? Or by recurrence $a_n=a_{n-1}(e^3)$? Or should I find a ...
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Understanding proof of sequential continuity?

I'm trying to understand proof of the following statement: Q. Let $f$ be a function on a closed bounded interval $[a,b]$. Prove that $f$ is continuous at $c \in [a,b]$ if and only if $f(x_n) \to c$...
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Probably dumb limit

I have a sequence of continuous functions $f_n : I^k \rightarrow I^k$ converging uniformly to a continuous function $f$. Then for each $n$ I choose a point $x_n$ and since they're chosen in $I^n$ ...
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Is that statement true or false?

Are following statements true or false ? If function $f$ is differentiable at $x_0$, then the sequence n.$( f(x_0+(1/n)) - f(x_0))_{n\in\mathbb{N}}$ is convergent. I am really not sure, but I think ...
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Does $\lim\limits_{x \rightarrow c} f(x)$ exist if the sequence $\{ f(x_n)\}_{n=1}^\infty$ is Cauchy?

I'm struggling a little with this question: Let c be a cluster point of $A ⊂ \mathbb{R}$, and $f : A → \mathbb{R}$ be a function. Suppose for every sequence $\{x_n \}$ in A, such that $\lim x_n = c$,...
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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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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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Show that $f_n(x)= \frac{\sin^2(n^{\alpha}x)}{nx}$ is not uniformly convergent for $\alpha \geq 1$ and $x \neq 0$

Can someone help me with proving that \begin{align} f_n: \mathbb{R} \rightarrow \mathbb{R }, \hspace{0.5cm} f_n(x)= \begin{cases} \frac{\sin^2(n^{\alpha}x)}{nx} & \text{ if } x \neq 0, \\ 0 &...
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Uniform convergence and the supremum theorem

I know that there are at least 3 other questions with the similar problem. I hope, though, you won't flag this as a duplicate since I've got some specific questions here that haven't been answered ...
Uniform convergence of $f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$
I need to study the uniform convergence of $$f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$$ on the interval $[2,+\infty)$ I've shown that on $[0,+\infty)$: at $x =0$ $f_n(0)=0 \xrightarrow{} 0$ at $x \neq 0$...