07/21/2012, 03:42 PM
I think the following will work to show convergeance
assume x , y and s to be reals > eta
1) the value of the product = 0 => done
2) the value is not zero => take the log => we get a sum
2b) show that the tail of the sequence goes to 0 faster than constant / n^2 hence we have convergeance. ( because of the famous Basel problem " zeta(2) = Pi^2 / 6 " )
As for the recursion , i dont know if it helps but remember
to go from s+1 to s do :
( we take the inverse with respect to y , notation : ^-1 )
f(x,y,s) = f(x,f^-1(x,y,s)+1,s)
which should be valid for all sufficiently large real non-integer s too.
regards
tommy1729
assume x , y and s to be reals > eta
1) the value of the product = 0 => done
2) the value is not zero => take the log => we get a sum
2b) show that the tail of the sequence goes to 0 faster than constant / n^2 hence we have convergeance. ( because of the famous Basel problem " zeta(2) = Pi^2 / 6 " )
As for the recursion , i dont know if it helps but remember
to go from s+1 to s do :
( we take the inverse with respect to y , notation : ^-1 )
f(x,y,s) = f(x,f^-1(x,y,s)+1,s)
which should be valid for all sufficiently large real non-integer s too.
regards
tommy1729
