## proof of convergence for the polygon circumscribing constant $\sum _{3} ^{\infty} \ln(\sec(\pi/ n))$

The polygon circumscribing constant is found by: $$\prod _3 ^\infty \sec \left( \frac\pi n \right)$$

I am trying to find a proof that this product converges. I know it is equal to:

$$\exp \left( \sum _{3} ^{\infty} \ln\left(\sec\left(\frac\pi n\right)\right) \right)$$

So I just need to show that sum converges. I do not see an easy way to use any of the convergence tests. In particular, I spent way to long trying to integrate this function to no avail.

So what convergence test is useable in this case?

## Limit of $n!(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!)$ as $n \to \infty$

$$L = \lim_{ n \to \infty} n!(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!) =0 \:$$

I know that $$\lim_{ n \to \infty}\sum_{k=0}^{n} 1/k! =e$$

So I’m assuming that $$L$$ goes to $$0$$ because $$(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!)$$ goes to $$0$$ faster than $$n!$$ goes to infinity. But how to prove this?

## Limits: factoring out x from $\lim _{x\to +\infty }\left(\frac{5-x^3}{8x+2}\right)$

So my teacher said that I cannot use arithmetic operation to factor out x from this type of equation, saying that it’s because it’s composed only by addition and subtraction. But I don’t understand clearly, because I get the right answer (according to the book):

$$\lim _{x\to +\infty }\left(\frac{5-x^3}{8x+2}\right) =\lim _{x\to \infty \:}\frac{x×\left(\frac{5}{x}-x^2\right)}{x×\left(8+\frac{2}{x}\right)}=\lim _{x\to \infty \:}\frac{\frac{5}{x}-x^2}{8+\frac{2}{x}}=\frac{\lim _{x\to \infty \:}\left(\frac{5}{x}-x^2\right)}{\lim _{x\to \infty \:}\left(8+\frac{2}{x}\right)}=\frac{\lim _{x\to \infty \:}\left(\frac{5}{x}\right)-\lim _{x\to \infty \:}\left(x^2\right)}{\lim _{x\to \infty \:}\left(8\right)+\lim _{x\to \infty \:}\left(\frac{2}{x}\right)}=\frac{0-\infty }{8+0}=\frac{-\infty \:}{8}$$

Applying the infinity property: $$\frac{-\infty }{-c}=\infty$$

$$=-\infty$$

Can someone explain to me why I can’t factor x out?

## Is it possible to find a sequence $(y_j)$ s.t. $\lim_{k \rightarrow \infty} \|a_k-y \|_{\ell^q}$ if $(a_k) \in \ell_p$?

Is it possible to find a sequence $$(y_j) \in \ell_p$$ s.t. $$\lim_{k \rightarrow \infty} \|a_k-y \|_{\ell^q}$$ if $$(a_k) \in \ell_p$$?

Why?

## Let $(X , \cal{A}, m)$ be a measure space. Let $f:X \to [0,1]$ be measurable. If $m(X) < \infty$, find$\lim_{n\to\infty} \int f^n \, d m$.

Let $$(X , \cal{A}, m)$$ be a measure space. Let $$f:X \to [0,1]$$ be a measurable function. If $$m(X) < \infty$$, determine $$\lim_{n\to\infty} \int f^n \, d m$$.

So far I have:

If $$f(x) < 1$$, then $$\lim_{n\to \infty}{f^n(x)} = 0$$. If $$f(x) = 1$$, then $$\lim_{n\to\infty}{f^n(x)} =1$$. So, for each $$x \in X$$, $$\lim_{n\to\infty}{f^n(x)} = \chi_{_{[f = 1]}}(x).$$

However, I am stuck because I cannot use the Lebesgue Monotone Convergence Theorem, since the sequence is decreasing. Also, I do not know where I will use the hypothesis that $$X$$ is a finite measure space. Any ideas?

## Let $a_n$ complex sequence prove that if $a_n\to \infty$ then $|a_n|\to\infty$. Note that $a_n = x_n + y_ni$

Let $$a_n$$ complex sequence prove that if $$a_n\to \infty$$ then $$|a_n|\to\infty$$. Note that $$a_n = x_n + y_ni$$ i dont know how to write that mathmatically.

trial :

Can i say that for every $$M>0$$ there exist $$N$$ such that for every $$n>N$$ ,

$$~~|x_n|>M~~ OR ~~~|y_n|>M$$ ( At least one of them goes to $$\infty$$)

because of that $$|an| = \sqrt{(x_n)^2+(y_n)^2} > M$$ and so $$|a_n|\to\infty$$.

## $\underset{n\rightarrow +\infty }{\overset{}{\lim }} \ \left(\sqrt[n]{2} -1\right)^{n} =0$

Prove that:

$$\underset{n\rightarrow +\infty }{\overset{}{\lim }} \ \left(\sqrt[n]{2} -1\right)^{n} =0$$

I would like a solution without integral, limit of real functions or others advanced methods. I thought $$\underset{n\rightarrow +\infty }{\overset{}{\lim }} \ 2\left(1- \frac{1}{\sqrt[n]2}\right)^{n} =0$$ but I don’t know how to continue.

## $\underset{n\rightarrow +\infty }{\overset{}{\lim }} \ \frac{10^{\sqrt{(\ln n)^{2} +\ln( n^{2}})}}{n^{2} +1} =+\infty$

Prove that:

$$\underset{n\rightarrow +\infty }{\overset{}{\lim }} \ \frac{10^{\sqrt{(\ln n)^{2} +\ln( n^{2}})}}{n^{2} +1} =+\infty$$

It is an exercise on first chapters of calculus textbook. I think it is possible to solve without integral or others advanced methods.

## When is $\lim_{n \rightarrow \infty} \mathbb{E}[X|\mathcal{F}_{t}] =\mathbb{E}[X|\mathcal{F}_{\infty}]$?

Is there a theorem like monotone convergence or dominated convergence for a problem of this sort?

## Smash product and the integers in a Grothendieck $(\infty, 1)$-topos

Let $$\mathcal{H}$$ be a Grothendieck $$(\infty,1)$$-topos. According to this page in nlab, for any $$X \in \mathcal{H}$$, the suspension object $$\Sigma X$$ is homotopy equivalent to the smash product $$B \mathbb{Z} \wedge X$$, where $$B \mathbb{Z}$$ is the “classifying space of the discrete group of integers.” Furthermore, for any pointed object $$X \in \mathcal{H}_*$$ and any group object $$G \in Grp(\mathcal{H})$$, the article says we can “form the tensor product $$X \otimes G \in Grp(\mathcal{H})$$.”

My problem is: none of this terminology is explained, nor does the page provide a reference. Specifically, what is $$\mathbb{Z}$$ in an arbitrary $$\infty$$-topos? What is the smash product $$\wedge$$? What is the tensor product $$\otimes$$? My best guess is that $$\otimes$$ refers to the unique tensor structure on $$\mathcal{H}_*$$ such that the map $$\mathcal{H} \to \mathcal{H}_*$$ is symmetric monoidal (here $$\mathcal{H}$$ is given the Cartesian monoidal structure), but this is only a guess.

Is there a reference where all these notions are defined?