Two Posed Problems Ready For Your Investigation....
I am passing on a problem-solving opportunity sent to me by L.W. (CA)...
To my knowledge, I am the originator of each and neither has been definitively solved. I refer to them as "Going off on a Tangent Problems" or "Shower Problems". The latter comes from the fact that many good problems, including these two, have occurred to me while standing in a hot shower.
[Question 1] Does the infinite product [tan(1)][tan(2)][tan(3)] ... converge? That is, does the sequence of partial products [tan(1)][tan(2)][tan(3)]...[tan(n)] converge? If not, is it bounded?
[Question 2] Consider natural number solutions to the inequality tan(n) > n. I've found three: 1, 260515 and 37362253. Are there more? Infinitely more? If there are infinitely many solutions, then would there also be infinitely many solutions to tan(n) > en? If so, name one.
Both problems lend themselves to experimental (computer) investigations. Mathematica (or some such software) may shed some light on things.
Please send me your insights and potential solutions, which I will both publish and pass on to L.W.
Hint: Sorry...none available
Solution Commentary: Sorry...none available...until someone sends me a solution or two....But see Archive entry for August 30, 2009.