2017-06-17 05:11:59 UTC
Leo Wapner asked in sci.math:
a(n) = tan(1) tan(2) ... tan(n)
does a(n) converge, or does it diverge?
Dec 28, 2006 , sci.math archives:
< http://mathforum.org/kb/message.jspa?messageID=5461260 > .
In 2009, Jim Ferry proved in sci.math that:
for any integers n,k > 0 with n even and k relatively prime to n, we have:
prod(X=1,n-1,tan(X*(k/n)*Pi/2) ) = (-1)^((k-1)/2)
and based on his proof, I sketched a proof that if n,k > 0 with n odd
and k odd and n relatively prime to n, then:
| prod(X=1,n-1,tan(X*(k/n)*Pi/2)) | = 1.
So, if n, k > 0 , with k odd and gcd(n,k) = 1, then:
| prod(X=1,n-1,tan(X*(k/n)*Pi/2)) | = 1 .
If , in the above, n/k is sufficiently close to Pi/2 ,
in other words if (k/n)*Pi/2 is sufficiently close to 1,
then | a(n) | is of the order of , well it depends how close
k*(Pi/2) is to n.... and k odd so tan( k *Pi/2) undefined (alias +/- oo ).
< http://mathforum.org/kb/message.jspa?messageID=6851966 > .
This motivates the question:
If xi is an irrational number, for what values of C>0 , if any,
can one be assured that there exist infinitely many co-prime integers
m, n with n>0, n odd, m positive or negative, such that:
| xi - m/n | < C/(n^2) ?
This has a resemblance to the question answered by Hurwitz's theorem in
< https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory) > ,
except here, we require the denominator n to be an *odd* number ...