The claim is $\exists$ N such that for $\it{all}$ k > N, $\frac{2 + Cos(k)}{\sqrt{k+1}}$ > $\frac{1}{\sqrt{k}}$. The inequality fails, for example, at k=22, k=355 and k=3126535, all of which are related to very good rational approximations of $\pi$ with (necessarily) odd denominators. It therefore boils down to whether the improvement in rational approximation to $\pi$ for larger k can effectively outpace the difference in reciprocal square-roots of k and k+1. Any ideas on an analytical approach welcome.
Real Analysis - Rational approximation of Pi
2
$\begingroup$
real-analysis
-
0I believe you are claiming that the irrationality measure of $\pi$ is $2$, that is a long-standing open problem. – 2017-02-09
-
0Thanks Jack - I've done some reading on the topic and suspect you may well be correct. – 2017-02-10