Wed Apr 16 20:38:42 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Products ---------- -- => [640 pi^3]/[2187 sqrt(3)] [Gradshteyn and Ryzhik 8.338(5)] product(Gamma(k/3), k = 1..8) _ 1 _ 2 _ 4 _ 5 _ 7 _ 8 (1) | (-)| (-)| (-)| (-)| (-)| (-) 3 3 3 3 3 3 Type: Expression Integer Time: 1.18 (IN) + 0.25 (EV) + 0.42 (OT) + 0.18 (GC) = 2.03 sec -- => n! = gamma(n + 1) product(k, k = 1..n) n ++-++ (2) | | k | | k= 1 Type: Expression Integer Time: 0.45 (IN) + 0.07 (OT) = 0.52 sec -- => x^[n (n + 1)/2] product(x**k, k = 1..n) n ++-++ k (3) | | x | | k= 1 Type: Expression Integer Time: 0.07 (IN) + 0.02 (EV) + 0.03 (OT) = 0.12 sec -- => n product((1 + 1/k), k = 1..n - 1) n - 1 ++-++ k + 1 (4) | | ----- | | k k= 1 Type: Expression Integer Time: 0.27 (IN) + 0.03 (EV) + 0.03 (OT) = 0.33 sec -- => 1/2^(2 n) binomial(2 n, n) [Knopp, p. 385] product((2*k - 1)/(2*k), k = 1..n) n ++-++ 2k - 1 (5) | | ------ | | 2k k= 1 Type: Expression Integer Time: 0.07 (IN) + 0.02 (EV) + 0.02 (OT) = 0.10 sec -- => [x^(2 n) - 1]/(x^2 - 1) [Gradshteyn and Ryzhik 1.396(1)] product(x**2 - 2*x*cos(k*%pi/n) + 1, k = 1..n - 1) n - 1 ++-++ k %pi 2 (6) | | - 2x cos(-----) + x + 1 | | n k= 1 Type: Expression Integer Time: 0.56 (IN) + 0.07 (EV) + 0.18 (OT) + 0.03 (GC) = 0.85 sec -- => 2/3 [Knopp, p. 228] limit(product((k**3 - 1)/(k**3 + 1), k = 2..n), n = %plusInfinity) (7) "failed" Type: Union("failed",...) Time: 0.45 (IN) + 0.27 (EV) + 0.09 (OT) + 0.03 (GC) = 0.83 sec -- => 2/pi [Gradshteyn and Ryzhik 0.262(2)] limit(product(1 - 1/(2*k)**2, k = 1..n), n = %plusInfinity) (8) "failed" Type: Union("failed",...) Time: 0.10 (IN) + 0.08 (EV) + 0.03 (OT) = 0.22 sec -- => sqrt(2) [Gradshteyn and Ryzhik 0.261] limit(product(1 + (-1)**(k + 1)/(2*k - 1), k = 1..n), n = %plusInfinity) (9) "failed" Type: Union("failed",...) Time: 0.08 (IN) + 0.17 (EV) = 0.25 sec -- => -1 [Knopp, p. 436] limit(product((k*(k + 1) + 1 + %i)/(k*(k + 1) + 1 - %i), k = 0..n), _ n = %plusInfinity) (10) "failed" Type: Union("failed",...) Time: 0.98 (IN) + 0.50 (EV) + 0.20 (OT) + 0.03 (GC) = 1.72 sec -- ---------- Quit ---------- )quit real 23.4 user 7.8 sys 0.5