Mathematica 3.0.0 bug in LerchPhi function

*To*: mathgroup at smc.vnet.net*Subject*: [mg9065] Mathematica 3.0.0 bug in LerchPhi function*From*: koehler at REMOVE-THIS.math.uni-bonn.de (Kai Koehler)*Date*: Thu, 9 Oct 1997 01:42:45 -0400*Organization*: RHRZ - University of Bonn (Germany)*Sender*: owner-wri-mathgroup at wolfram.com

In article <6171ii$osf at smc.vnet.net>, luca ciotti <ciotti at boas5.bo.astro.it> wrote: > I found a problem in symbolic summation using Mathematica 3.0.0. > > A = Sum[ Exp[-k]/(1+k^3), {k,0,Infinity}] > > The analitycal result, when evaluated numerically is > > N[A,50] = 0.5344386..... > > > This is obviously wrong, in fact each term of the series is > positive, and the first (k=0) is equal 1. > > Performing directly NSum, return 1.20111..... Hi, I verified the exact symbolic result which is given by Mathematica and it is correct (albeit written up in a too complicated way, even after Simplify...). When you write it as (2 + (-1)^(2/3)/E*LerchPhi[1/E, 1, (-1)^(1/3)] - (-1)^(1/3)/E*LerchPhi[1/E, 1, -(-1)^(2/3)] - E*Log[1 - 1/E])/3 then even Mathematica is able to evaluate it correctly. The actual problem is a major bug in the numerical calculation of the LerchPhi function: Try e.g. z = 0.4; v = -0.5; Plot[NSum[z^n/(n + v), {n, 0, 20}] - LerchPhi[z, 1, v], {v, -1, 1}] The numerical sum is the truncated definition of the LerchPhi function (which converges very well). For negativ v you shall see a huge error. It took me some time to find this out. Isn't it just wonderfull that we serve as beta testers for Wolfram and we shall never get a free bug fix (if there shall ever be any bug fix at all...)? Imagine that I paid money for this super-buggy product. Best regards Kai Koehler