Thu Feb 25 19:38:49 MST 1999 aquarius% maple |\^/| Maple V Release 5.1 (WMI Campus Wide License) ._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. # ----------[ M a p l e ]---------- #interface(echo = 3); # ---------- Initialization ---------- > readlib(showtime): > on; # ---------- Products ---------- # => [640 pi^3]/[2187 sqrt(3)] [Gradshteyn and Ryzhik 8.338(5)] O1 := product(GAMMA(k/3), k = 1..8); 640 3 1/2 ---- Pi 3 6561 time = 0.04, bytes = 67082 # => n! = gamma(n + 1) O2 := Product(k, k = 1..n); n --------' ' | | | | k | | | | k = 1 time = 0.01, bytes = 9494 O3 := value(%); GAMMA(n + 1) time = 0.02, bytes = 40586 # => x^[n (n + 1)/2] O4 := product(x^k, k = 1..n); n --------' ' | | k | | x | | | | k = 1 time = 0.38, bytes = 541666 # => n O5 := product((1 + 1/k), k = 1..n - 1); GAMMA(n + 1) ------------ GAMMA(n) time = 0.02, bytes = 20518 O6 := simplify(%); n time = 0.06, bytes = 61638 # => 1/2^(2 n) binomial(2 n, n) [Knopp, p. 385] O7 := product((2*k - 1)/(2*k), k = 1..n); (n + 1) (n + 1) (1/2) 2 GAMMA(n + 1/2) ------------------------------------ 1/2 GAMMA(n + 1) Pi time = 0.04, bytes = 43414 O8 := simplify(%); bytes used=1000072, alloc=786288, time=0.73 GAMMA(n + 1/2) ------------------ 1/2 GAMMA(n + 1) Pi time = 0.28, bytes = 259370 # => [x^(2 n) - 1]/(x^2 - 1) [Gradshteyn and Ryzhik 1.396(1)] O9 := Product(x^2 - 2*x*cos(k*Pi/n) + 1, k = 1..n - 1); n - 1 --------' ' | | / 2 k Pi \ | | |x - 2 x cos(----) + 1| | | \ n / | | k = 1 time = 0.02, bytes = 23138 O10 := value(%); n - 1 --------' ' | | / 2 k Pi \ | | |x - 2 x cos(----) + 1| | | \ n / | | k = 1 time = 0.33, bytes = 467682 # => 2/3 [Knopp, p. 228] O11 := product((k^3 - 1)/(k^3 + 1), k = 2..infinity); bytes used=2000632, alloc=1310480, time=1.60 bytes used=3000872, alloc=1769148, time=2.79 bytes used=4001124, alloc=2031244, time=4.13 bytes used=5003080, alloc=2293340, time=5.47 2/3 time = 4.29, bytes = 3361226 # => 2/pi [Gradshteyn and Ryzhik 0.262(2)] O12 := product(1 - 1/(2*k)^2, k = 1..infinity); bytes used=6003288, alloc=2358864, time=6.94 2 ---- Pi time = 1.75, bytes = 1191290 # => sqrt(2) [Gradshteyn and Ryzhik 0.261] O13 := Product(1 + (-1)^(k + 1)/(2*k - 1), k = 1..infinity); infinity --------' / (k + 1)\ ' | | | (-1) | | | |1 + -----------| | | \ 2 k - 1 / | | k = 1 time = 0.01, bytes = 7250 O14 := value(%); bytes used=7003468, alloc=2358864, time=8.20 product: "Cannot show that 1+(-1)^(k+1)/(2*k-1) has no zeros on [1,infinity]" infinity --------' / (k + 1)\ ' | | | (-1) | | | |1 + -----------| | | \ 2 k - 1 / | | k = 1 time = 1.21, bytes = 859986 # => -1 [Knopp, p. 436] O15 := Product((k*(k + 1) + 1 + I)/(k*(k + 1) + 1 - I), k = 0..infinity); infinity --------' ' | | k (k + 1) + 1 + I | | ----------------- | | k (k + 1) + 1 - I | | k = 0 time = 0.01, bytes = 7206 O16 := value(%); product: "Cannot show that (k*(k+1)+1+I)/(k*(k+1)+1-I) is continuous on [0,infinity]" infinity --------' ' | | k (k + 1) + 1 + I | | ----------------- | | k (k + 1) + 1 - I | | k = 0 time = 0.66, bytes = 583142 # ---------- Quit ---------- O17 := quit bytes used=7813564, alloc=2555436, time=9.23 real 10.56 user 9.30 sys 1.17