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Is sin(n!) dense in [0.1, 1] ?

Anonymous Fri 10 Dec 13:01:54 2021 No.13946448 View Reply Original Report

Quoted By: >>13946750 >>13948599

Hi /sci/,

Do you know if the sequence $\sin (n!)$ is dense in $[0.1, 1]$ ? With a proof.

I can't even show that

$\sin (n!)$ does not converge.

Anonymous

Anonymous Fri 10 Dec 2021 13:45:39 No.13946528 Report

Quoted By: >>13946650

sin(x!) converges no more than sin(x) converges.

I don't know in what sense you're using the word sequence or dense, since you're allowing for a rational index.

Anonymous

Anonymous Fri 10 Dec 2021 14:40:36 No.13946650 Report

Quoted By:

>>13946528

density has a unique meaning. Is it true that for every $x \in [-1, 1]$ there is a sequence of integers $n_1, ..., n_k, ....$ such that $\sin (n_k ! ) \leftarrow x$ when k approaches infinity ?

Anonymous

Anonymous Fri 10 Dec 2021 15:31:51 No.13946750 Report

Quoted By:

>>13946448
I don't know how to prove this but this looks useful: https://math.stackexchange.com/a/743392

Anonymous

Anonymous Fri 10 Dec 2021 15:39:37 No.13946780 Report

Quoted By: >>13946783 >>13946795

The answer will be yes if $n!$ is dense in $(0,2\pi)$ . I will think about if this is true.

Anonymous

Anonymous Fri 10 Dec 2021 15:40:41 No.13946783 Report

Quoted By:

>>13946780
sorry I mean n! modulo 2 pi of course

Anonymous

Anonymous Fri 10 Dec 2021 15:45:31 No.13946795 Report

Quoted By: >>13946821

>>13946780
might be true for all irrational numbers. n modulo x is dense in (0,x) for all irrationak x, pretty sure. Can we extend this to only n!?

Anonymous

Anonymous Fri 10 Dec 2021 15:51:28 No.13946821 Report

Quoted By:

>>13946795
The following should be noted:
Suppose (n - 1)! = k x + r x for k integer and r in (0,1). Then n! = k n x + r n x, so basically
n! mod x = n * ((n-1)! mod x) mod x.
So the factorial sequence somehow commutes with the modulo operation.

Anonymous

Anonymous Fri 10 Dec 2021 15:55:55 No.13946835 Report

Quoted By: >>13947085

shut the fuck up nerds

Anonymous

Anonymous Fri 10 Dec 2021 17:28:38 No.13947085 Report

Quoted By:

>>13946835
Sir this is a science & math board

Anonymous

Anonymous Sat 11 Dec 2021 00:39:54 No.13948599 Report

Quoted By:

>>13946448
https://math.stackexchange.com/questions/8690/is-there-a-limit-of-cosn

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