[math-fun] nice product for (1+y)^(1/a)

4 Jun 2003

      I like this one:

Let 

                                       y                  
                   Y =y      T(y)  = + -                                   
                    1                  a                  

and 

                              a       a             a                  
                      (1+Y ) a -(a+Y )      (1+Y ) a                   
                          k         k           k                  
                 Y   =-----------------  =  --------- - 1                  
                  k+1            a                 a                   
                           (a+Y )            (a+Y )                    
                               k                 k                   

then
     ___oo                  1/a
     | |   (1+T(Y )) = (1+y)   
     | |k=1      k

Convergence is quadratic.

The ansatz (T(y)="terms 1..(n-1) of Taylor series") also works for
1/\sqrt[a]{1+y} and gives products with n-th order of convergence.

-- 
p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0
Life is hard and then you die.

[math-fun] nice product for (1+y)^(1/a)

Joerg Arndt