- Thread starter
- #51

#### chisigma

##### Well-known member

- Feb 13, 2012

- 1,704

From the expression of the function sin x as infinite product ...Posted the 08 16 2014 on www.mathhelpforum.com by the user superzhangmch and not yet solved...

... prove...

$\displaystyle \lim_{n \rightarrow \infty} \frac{\ln |\sin n|}{n} = 0\ (1)$

...or show it is not true...

$\displaystyle \sin x = x\ \prod_{k=1}^{\infty} (1 - \frac{x^{2}}{k^{2}\ \pi^{2}})\ (1)$

… we derive...

$\displaystyle a_{n} = \frac{\ln |\sin n|}{n} = \frac{\ln n}{n} +\sum_{k=1}^{\infty} \frac{\ln |1 - \frac{n^{2}}{k^{2}\ \pi^{2}}|}{n}\ (2)$

The proposed question is not trivial since to show that $\displaystyle \lim_{n \rightarrow \infty} a_{n} = 0$ it is necessary to show that each term of the series (2) tends to zero as n tends to infinity. This can be critical when it is $\displaystyle \frac{n}{k} \sim \pi$, that is when $\displaystyle \frac{n}{k}$ is a 'good approximation' of $\pi$ since the logarithm can take negative values even higher. The workload needed for this investigation is not light but fortunately with a short research it has found a German text of the late nineteenth century, where are example values $\displaystyle \frac{n}{k}$ 'good approximations' of $\pi$...

*Archimedes,Huygens, Lambert, Legendre.*

Vier Abhandlungen über die Kreismessung. Deutsh hrsg. und mit einerÜbersicht über die

Geschichte des Problemes von der Quadratur des Zirkels

Vier Abhandlungen über die Kreismessung. Deutsh hrsg. und mit einerÜbersicht über die

Geschichte des Problemes von der Quadratur des Zirkels

Published 1892 by B.G. Teubner in Leipzig .

Written in German.

pages 146-147 has a table of rational approximations of pi...

1:3

7:22

106:333

113 : 355

33102: 103993

33215 : 104348

66317: 208341

99532 : 312689

265381: 833719

364913 : 1146408

1360120: 4272943

1725033 : 5419351

25510582: 80143857

52746197 : 165707065

78256779: 245850922

131002976 :411557987

340262713 :1068966896

811528438 : 2549491779

1963319607 : 6167950454

4738167652: 14885392687

6701487259 : 21053343141

567663097408 : 1783366216531

1142027682075 : 3587785776203

1709690779483 : 5371151992734

2851718461558 : 8958937768937

107223273857129 : 336851849443403

324521540032945 : 1019514486099146

The next job in the next posted ...

Kind regards

$\chi$ $\sigma$