# Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use the tag “limits-colimits” instead.

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### Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
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### An iterative logarithmic transformation of a power series

Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion: $$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$ Then, at each step ...
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### The limit of $(\sin(n!)+1)^{1/n}$ as n approaches infinity

Calculate the limit $$\lim_{n\rightarrow\infty}(\sin(n!)+1)^{1/n}$$ or prove that the limit does not exist. This appeared as a problem in my mathematical analysis test, and the answer was that the ...
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### Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
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### Study the convergence of $\sum_{n=1}^{\infty}\frac{\sin (n\sqrt{n})}{\sqrt{n}}.$

I try to use the following result: If $f(x)\in C^1[1,+\infty)$ and $\displaystyle\int_1^{+\infty} |f'(x)|{\rm d}x$ is convergent, then $\displaystyle\sum_{n=1}^{\infty} f(n)$ has the same convergence ...
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I am trying to prove, for the general case whereby $\zeta(\cdot\,,\cdot)$ is the Hurwitz-Zeta function, and $a\in \mathbb{N}$, that $$\mathcal{L} = \lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\... 6 votes 1 answer 229 views ### Density of \{\sin(x^n)|n\in\mathbb{N}\} for x>1 While reading other topics, e,g, Is n \sin n dense on the real line? or Is \{ \sin n^m \mid n \in \mathbb{N} \} dense in [-1,1] for every natural number m?, the following problem appeared in ... • 3,994 6 votes 0 answers 788 views ### Weird use of Glasser's Master Theorem Consider the following enumeration of the rational numbers in [\,0,1):$$0, \frac{1}{2},\frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{...
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Let $f(x) = \sum_{n=1}^\infty a_n x^n$. What is $\lim_{x\rightarrow \infty} f(x)$ in terms of the $a_i$? That question may be too broad, so here are some restrictions: Assume f(x) is continuous (and ...
Consider the following sequence of operations : Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of $n-2$...