[MUG] bug in sum?

[MUG] Re: bug in sum?
Author: Maple User Group    Posted: Tue, 28 May 2002 18:50:06 -0400
>> From: Maple User Group "maple_gr" -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Mon, 27 May 2002 09:55:20 +0400 From: Vladimir Bondarenko "vvb" To: "maple-list" Subject: bug in sum? Dear Jorn Justiz, You identified a BUG. > kernelopts(version); `Maple 7.00, IBM INTEL NT, May 28 2001 Build ID 96223` > sum((k+1)*a^k, k=2..infinity): evalf(subs(a=1/2, %)); 12. > sum((k+1)*(1/2)^k, k=2..infinity); 2 This bug is absent in Maple V, Release 5 > kernelopts(version); Maple V, Release 5, IBM INTEL NT, Nov 27 1997, WIN-55NC-567980-7 > sum((k+1)*a^k,k=2..infinity): evalf(subs(a=1/2, %)); 2. Best wishes, Vladimir Bondarenko Applied mathematician -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Mon, 27 May 2002 15:15:07 +0200 From: Preben Alsholm "ifakpa" To: "maple-list" "max_joern" Subject: bug in sum? I think that the expression with LegendreP is OK, but think that the problem lies in the procedure `evalf/LegendreP`. In that procedure's statement 18 all values of LegendreP(v,u,z) having integral values of v and u and having v<abs(u) are set to 0. >>From Abramowitz & Stegun 8.1.2 we find that LegendreP(1, -3, (1+x)/(1-x)) = (1/6)*x^(3/2)*hypergeom([-1,2], [4], -x/(1-x) ). Maple itself simplifies the right hand side to 1/12*x^(3/2)*(x-2)/(x-1). If you in the output from sum((k+1)*x^k,k=3..infinity); replace LegendreP(...) with this expression, then the sum-result is seen to be correct. If I'm correct in claiming that there is a bug in `evalf/LegendreP` then the bug is present in Maple V, 5.1, Maple 6, Maple 7, and unfortunately also in Maple 8. In Maple V, 5.1 and Maple 6 your problem with the sum doesn't show up since the result doesn't contain LegendreP. -- Preben Alsholm Institut for matematik (Department of Mathematics) DTU (Technical University of Denmark) | >> From: "max_joern" | | I recently stumbled over this sum: (I'm using Maple 7) | | Sum((k+1)*ee^k,k = 3 .. infinity). This should converge for | abs(ee)<1. We know that the sum from 0 to infinity equals in that | case 1/(1-ee)^2 which is checked by maple: | | > sum((k+1)*(ee)^k,k=0..infinity); | 1/((ee-1)^2) | | (no convergence check b.t.w.) | | If we plug in ee=-0.5 we obtain: | > subs(ee=-0.5,%); | 0.4444444 | | Now Sum((k+1)*ee^k,k = 3 .. infinity) should be Sum((k+1)*ee^k,k = 0 | .. infinity) - Sum((k+1)*ee^k,k = 0 .. 2), right? This we can | calculate for the case ee=-0.5 to be equal to 0.4444444-0.75= | -.3055555556. But here the surprise: if we use maple to calculate the | sum from 3 to infinity with arbitrary value of ee first and then | substitute ee=-0.5 we get: | > sum((k+1)*(ee)^k,k=3..infinity); | 4*ee^3*(-9/ee^(1/2)/(-ee+1)^2*LegendreP(1,-3,(ee+1)/(-ee+1))-3/2*1/(ee-1)^5*(-ee+1)^2/ee+1/(ee-1)^5*(-ee+1)^2/ee^2-1/4*1/(ee-1)^5*(-ee+1)^2/ee^3-1/4*1/ee^3*(-ee+1)) | | > evalf(subs(ee=-0.5,%)); | -.1666666667-0.*I | | which is clearly different from the expected result. I assume that | already the expression with the associated Legendre functions is wrong | but I'm not sure. We get the expected result though if we take the | limit: | > sum((k+1)*(ee)^k,k=3..n); | ee^(n+1)*(-2+(n+1)*ee-n)/(ee-1)^2-ee^3*(-4+3*ee)/(ee-1)^2 | | >limit(subs(ee=-0.5,%),n=infinity); | -.3055555556 |

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