Re: [Maxima-discuss] simplify_sum leads to a wrong value

[philippe <rou...@gm...>]

Hi,

3 years after our discussion about simplify_sum errors, I've found 
another example where maxima (and sagemath) fails to simplify a 
summation. Wolframalpha.com also fails to find the simplest expression 
(pi/16) but gives different closed forms and a numerical value 
compatible with it. Here is a minimal example showing the problem  in 
maxima :

'S=S:sum(sin(n*%pi/3)^3*cos(n*%pi/3)/n,n,1,inf);
'S=S:ratsimp(simplify_sum(S2)); /* false!*/
S_exact:%pi/16;
S_num:float(sum(sin(n*%pi/3)^3*cos(n*%pi/3)/n,n,1,10000));
S_exact=float(S_exact);

you can check that the value given by simplify_sum is far from the 
numerical value, and this one is compatible with pi/16 (which has been 
obtained by Dirichlet Theorem for the Fourier series of some  piecewise 
constant function):

S= 3/8*sqrt(3)*e + 3/8*sqrt(3) = 2.4150948914066888
S_exact= 1/16*pi = 0.19634954084936207
S_num= 0.1965659389111564

I've added this example in comment to bug report 3630 :

https://sourceforge.net/p/maxima/bugs/3630/

thanks for reading,
Philippe

Le 07/04/2020 à 22:52, philippe a écrit :
> Hi,
> 
> I want to check the limit of the following serie
> 
> sum((k*sin((k*%pi)/3))/(4*k^2-1),k,1,inf)
> 
> I know it converge to (sqrt(3)*%pi)/16=0.3400873807939158...
> 
> simplify_sum find the limit in terms of erf(%i) but floating point
> approximation is far from the expected value : 2.564242426927912...
> 
> I don't know if the problem comes from simplify_sum or floating point
> approximations of erf function, but there is no error for the similar serie
> 
> sum(k*sin(k*%pi/2)/(4*k^2-1),k,1,inf)=sqrt(2)*%pi/16
> 
> if some one is interested in this problem here is a short piece of code
> to test it :
> 
> load(simplify_sum)$
> S2:sum(k*sin(k*%pi/3)/(4*k^2-1),k,1,inf)$
> S2s:simplify_sum(S2)$
> S2c:sqrt(3)*%pi/16$
> print(S2,"=",S2s,"=",float(S2s));
> print(S2,"=",S2c,"=",float(S2c));
> 
> thanks for reading,
> Philippe