Diophantine Approximations to the Constant, ln(2),

    in terms of ratios of sequences satisfying the same recurrence



                              By Shalosh B. Ekhad



       Let a(n), b(n) be the sequences satisfying the linear recurrence



           (n + 1) x(n) + (-6 n - 9) x(n + 1) + (n + 2) x(n + 2) = 0



                              and in Maple format



(n+1)*x(n)+(-6*n-9)*x(n+1)+(n+2)*x(n+2) = 0


                       subject to the initial conditions



                               a(1) = 2, a(2) = 9





                              b(1) = 3, b(2) = 13



 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, ln(2)



The delta such that the error is less than the reciporocal of the denominato\

    r to the power delta+1 is approximately, 0.2820366556



This indicated a proof that, ln(2),

    is irrational with an irrationality measure about, 4.545638413



                            -----------------------





                                              2
                                            Pi
Diophantine Approximations to the Constant, ---,
                                             6

    in terms of ratios of sequences satisfying the same recurrence



                              By Shalosh B. Ekhad



       Let a(n), b(n) be the sequences satisfying the linear recurrence



             2              2                                2
     -(n + 1)  x(n) + (-11 n  - 33 n - 25) x(n + 1) + (n + 2)  x(n + 2) = 0



                              and in Maple format



-(n+1)^2*x(n)+(-11*n^2-33*n-25)*x(n+1)+(n+2)^2*x(n+2) = 0


                       subject to the initial conditions



                             a(1) = 5, a(2) = 125/4





                              b(1) = 3, b(2) = 19



                                                                            2
                                                                          Pi
  Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, ---
                                                                           6



The delta such that the error is less than the reciporocal of the denominato\

    r to the power delta+1 is approximately, 0.09900670825



                               2
                             Pi
This indicated a proof that, ---,
                              6

    is irrational with an irrationality measure about, 11.10032570



                            -----------------------





Diophantine Approximations to the Constant, Zeta(3),

    in terms of ratios of sequences satisfying the same recurrence



                              By Shalosh B. Ekhad



       Let a(n), b(n) be the sequences satisfying the linear recurrence



        3             2                                          3
 (n + 1)  x(n) - (17 n  + 51 n + 39) (2 n + 3) x(n + 1) + (n + 2)  x(n + 2) = 0



                              and in Maple format



(n+1)^3*x(n)-(17*n^2+51*n+39)*(2*n+3)*x(n+1)+(n+2)^3*x(n+2) = 0


                       subject to the initial conditions



                             a(1) = 6, a(2) = 351/4





                              b(1) = 5, b(2) = 73



Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, Zeta(3)



The delta such that the error is less than the reciporocal of the denominato\

    r to the power delta+1 is approximately, 0.08127423128



This indicated a proof that, Zeta(3),

    is irrational with an irrationality measure about, 13.30402286



                          ----------------------------



                          This took, 3.105, seconds.