so what is the average of the sequence made from cos(2)?
$\cos(2)=0.4161468365471423869975682295007621897660...$
the first number in the sequence is the first number in the decimal expansion until it hits a $0$ so $f(1)=4161468365471423869975682295$
The next number is the next number in the expansion
$f(2)=762189766$
$f(3)=771$
$f(4)=7554489$
You start the next number every time there is a $0$
If $\cos(2)$ is a normal number $0$ should be $10$% of the number expansion
My question is what's the average of $f(1),f(2),f(3),f(4),f(5),...$ or what is the limit of $(f(1)\times f(2)\times f(3)\times f(4)\times f(5)\times f(6)\times ...\times f(n))^{1/n}$ as n goes to infinty.
big numbers a rarer than small numbers witch should make it possible to converge.
For the fun of it, I made some calculations $$\left( \begin{array}{ccc} p & \text{arithmetic mean} & \text{geometric mean} \\ 50 & 9.3630\times 10^{51} & 5.7203\times 10^{11} \\ 100 & 4.6815\times 10^{51} & 7.4872\times 10^{10} \\ 150 & 3.1214\times 10^{51} & 1.5380\times 10^{11} \\ 200 & 2.3410\times 10^{51} & 6.5607\times 10^{10} \\ 250 & 1.8728\times 10^{51} & 5.7707\times 10^{10} \\ 300 & 2.8256\times 10^{61} & 1.2065\times 10^{11} \end{array} \right)$$