David Bernier

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 ).

cf.:

< 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

Diophantine approximation,

< https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory) > ,

except here, we require the denominator n to be an *odd* number ...

David Bernier

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 ).

cf.:

< 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

Diophantine approximation,

< https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory) > ,

except here, we require the denominator n to be an *odd* number ...

David Bernier