simpsum in maxima missing basic infinite series identities for : sum(k * x^k, 0, inf)

Question

Maxima's simpsum seems to be lacking some basic identities for infinite series. It has :

sum(x^k, k, 0, inf), simpsum=true;

but doesn't even have identities for:

sum(k * x^k, k, 0, inf), simpsum = true;

Is there a more comprehensive series summation in maxima beyond simpsum=true?

Answer 1

Take a look at simplify_sum which applies several kinds of identities, including Gosper's and Zeilberger's methods.

(%i1) load (simplify_sum);
(%o1)      /usr/share/maxima/5.35.1/share/solve_rec/simplify_sum.mac
(%i2) 'sum(k * x^k, k, 0, inf);
                                  inf
                                  ====
                                  \        k
(%o2)                              >    k x
                                  /
                                  ====
                                  k = 0
(%i3) simplify_sum (%);
                                      x
(%o3)                            ------------
                                  2
                                 x  - 2 x + 1