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8

\begin{align} P &=\large\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}\\ &=\large\lim_{m\to\infty}\prod_{n=1}^m e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}\\ &=\large\lim_{m\to\infty}e^m\left[\left(\frac 12\right)^1\left(\frac 23\right)^2\left(\frac 34\right)^3\cdots \left(\frac m{m+1}\right)^m\right] ... 5 We have:\log P = \sum_{n=1}^{+\infty}\left(1+\left(n+\frac{1}{2}\right)\log\left(1-\frac{1}{n+1}\right)\right)$$but:$$\sum_{n=1}^{N}\left(1+n\log n-(n+1)\log(n+1)\right) = N-(N+1)\log(N+1)$$since we have a telescopic sum, while:$$\sum_{n=1}^{N}\left(\frac{1}{2}\log n+\frac{1}{2}\log(n+1)\right)=\log(N!)+\frac{1}{2}\log(N+1)$$so: ... 5 The series is convergent since it is absolutely convergent. Since the sequence \{\{n/\pi\}\}_{n\in\mathbb{N}} is equidistributed \pmod 1, we have that |\sin n|\leq\frac{\sqrt{3}}{2} holds for about \frac{2N}{3} integers in the range [1,N], assuming that N is large enough. This gives:$$\left|\prod_{n=1}^{N}\sin n\right|\leq ...

4

$|a_{n+2}/a_n|$ is bounded above by a number $x<1$, so $$\sum_n |a_n|<C\sum_n x^{n/2}$$

3

From a simple google search, many teaching resources use the term "Telescoping Sums and Products": www.math.cmu.edu/.../3-telescope-solns.pdf (broken now, but from a reliable CMU edu website. Google says it is titled: III. Telescoping Sums and Products) http://faculty.wwu.edu/sarkara/ph13.pdf (Title of paper is "Telescoping Sums and Products") ...

2

No. Diamond and Pintz proved that for some constant $c>0$, there are arbitrarily large values of $x$ for which $$e^\gamma \log x \prod_{p\le x} \frac{p-1}p < 1 - \frac{c\log\log\log x}{x^{1/2}\log x}.$$ Lamzouri conjectures that the right-hand side can be improved to $$1 - \frac{(\log\log\log x)^2}{(2\pi+\varepsilon) x^{1/2}\log x}$$ but no further.

2

It is a special case of the Brunn-Minkowski Theorem. It isn't strange at all; it is a rather useful inequality in measure theory.

1

This is Mahler's inequality. We had it here in this question (see also the linked questions), and it was question A2 on the 2003 Putnam, so you can find some proofs in Kedlaya's archive. As for its usage: It is sometimes used to prove the Brunn-Minkowski inequality (of which it is a special case, as Umberto P. noted). In that proof, one first proves this ...

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