Tue Feb 23 11:50:39 MST 1999 aquarius% macsyma Starting Macsyma math engine with no window system... /opt/local/macsyma_422/macsyma.422 local This is Macsyma 422.0 for Sparc (Solaris 2.x) computers. Copyright (c) 1982 - 1998 Macsyma Inc. All rights reserved. Portions copyright (c) 1982 Massachusetts Institute of Technology. All rights reserved. Type "DESCRIBE(TRADE_SECRET);" to see important legal notices. Type "HELP();" for more information. /aquarius/data2/opt/local/macsyma_422/system/init.lsp being loaded. /aquarius/home/wester/macsyma-init.lsp being loaded. (c1) (c2) /* ----------[ M a c s y m a ]---------- */ /* ---------- Initialization ---------- */ symbol_display_case: lower_case$ Time= 0 msecs (c3) showtime: all$ Time= 0 msecs (c4) prederror: false$ Time= 0 msecs (c5) /* ---------- Limits ---------- */ /* Start with a famous example => e */ 'limit((1 + 1/n)^n, n, inf); Time= 0 msecs 1 n (d5) limit (- + 1) n -> inf n (c6) ev(%, limit); Time= 210 msecs (d6) %e (c7) /* => 1/2 */ limit((1 - cos(x))/x^2, x, 0); Time= 280 msecs 1 (d7) - 2 (c8) /* See Dominik Gruntz, _On Computing Limits in a Symbolic Manipulation System_, Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 1996. => 5 */ limit((3^x + 5^x)^(1/x), x, inf); Time= 3250 msecs (d8) 5 (c9) /* => 1 */ limit(log(x)/(log(x) + sin(x)), x, inf); Time= 40 msecs (d9) 1 (c10) /* => - e^2 [Gruntz] */ 'limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x^2/(x + 1)))) - exp(x))/x, x, inf); Time= 10 msecs - x %e x ----------------- 2 2 x - ----- x + 1 - x %e + %e x %e - %e (d10) limit ------------------------- x -> inf x (c11) ev(%, limit); Time= 24220 msecs - x %e x ----------------- 2 2 x - ----- x + 1 - x %e + %e x %e %e (d11) limit (------------------- - ---) x -> inf x x (c12) /* => 1/3 [Gruntz] */ 'limit(x*log(x)*log(x*exp(x) - x^2)^2/log(log(x^2 + 2*exp(exp(3*x^3*log(x))))), x, inf); Time= 440 msecs 2 x 2 x log(x) log (x %e - x ) (d12) limit -------------------------------- x -> inf 3 3 x log(x) %e 2 log(log(2 %e + x )) (c13) ev(%, limit); Time= 208640 msecs 2 x 2 x log(x) log (x %e - x ) (d13) limit -------------------------------- x -> inf 3 3 x log(x) %e 2 log(log(2 %e + x )) (c14) /* => 1/e [Knopp, p. 73] */ limit(1/n * n!^(1/n), n, inf); /aquarius/data2/opt/local/macsyma_422/library1/combin.so being loaded. Time= 880 msecs 1 (d14) -- %e (c15) /* Rewrite the above problem slightly => 1/e */ limit(1/n * gamma(n + 1)^(1/n), n, inf); Time= 740 msecs 1 (d15) -- %e (c16) /* => 1 [Gradshteyn and Ryzhik 8.328(2)] */ assume(a > 0)$ Time= 0 msecs (c17) limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, inf); Time= 780 msecs (d17) 1 (c18) forget(a > 0)$ Time= 0 msecs (c19) assume(a < 0)$ Time= 0 msecs (c20) limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, inf); Time= 1000 msecs (d20) 1 (c21) forget(a < 0)$ Time= 10 msecs (c22) /* => e^z [Gradshteyn and Ryzhik 9.121(8)] */ limit(hgfred([1, k], [1], z/k), k, inf); /aquarius/data2/opt/local/macsyma_422/library1/hyp.so being loaded. Time= 810 msecs z (d22) %e (c23) /* => Euler's_constant [Gradshteyn and Ryzhik 9.536] */ limit(zeta(x) - 1/(x - 1), x, 1); Time= 140 msecs (d23) infinity (c24) /* => gamma(x) [Knopp, p. 385] */ 'limit(n^x/(x * 'product((1 + x/k), k, 1, n)), n, inf); Time= 90 msecs x n limit ------------- n -> inf n /===\ | | x | | (- + 1) | | k k = 1 (d24) ---------------------- x (c25) q: closedform(%); /aquarius/data2/opt/local/macsyma_422/share/nusum1.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/binoml.so being loaded. Time= 1370 msecs x n n! x! (limit --------) n -> inf (x + n)! (d25) ---------------------- x (c26) assume(x > 0)$ Time= 0 msecs (c27) ev(q, 'limit); Time= 1190 msecs (d27) 0 (c28) forget(x > 0)$ Time= 0 msecs (c29) remvalue(q)$ Time= 0 msecs (c30) /* See Angus E. Taylor and W. Robert Mann, _Advanced Calculus_, Second Edition, Xerox College Publishing, 1972, p. 125 => 1 */ limit(x * integrate(exp(-t^2), t, 0, x)/(1 - exp(-x^2)), x, 0); Time= 300 msecs (d30) 1 (c31) /* => [-1, 1] */ [limit(x/abs(x), x, 0, minus), limit(x/abs(x), x, 0, plus)]; Time= 20 msecs (d31) [- 1, 1] (c32) /* => pi/2 [Richard Q. Chen] */ limit(atan(-log(x)), x, 0, plus); Time= 20 msecs %pi (d32) --- 2 (c33) /* ---------- Quit ---------- */ quit(); Bye. real 250.75 user 245.08 sys 1.72