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Listen up.
There are certain things that you should learn at some point during university, at least if you do a maths degree. This is one of them.
You know what continued fractions are. They are representations of real numbers that look like a0 + 1/(a1 + 1/(a2 + 1/(a3 + 1/...))). Ramanujan loved them. They pop up in MATH4104 lectures. You can get rational approximations of pi with them.
Anyway, consider the geometric mean of the a_j from j = 1 to n (ie, ignoring the integer component a0): (a1*a2*...*a_n)1/n.
Then consider the limit of this geometric mean as n goes to infinity. For almost all real numbers, this limit is a constant equal to about 2.685. It's called Khinchin's constant.
This is the craziest thing in the whole of analysis.
You know what continued fractions are. They are representations of real numbers that look like a0 + 1/(a1 + 1/(a2 + 1/(a3 + 1/...))). Ramanujan loved them. They pop up in MATH4104 lectures. You can get rational approximations of pi with them.
Anyway, consider the geometric mean of the a_j from j = 1 to n (ie, ignoring the integer component a0): (a1*a2*...*a_n)1/n.
Then consider the limit of this geometric mean as n goes to infinity. For almost all real numbers, this limit is a constant equal to about 2.685. It's called Khinchin's constant.
This is the craziest thing in the whole of analysis.