O17-C018
> restart;
> u:=n->x^n/(n+1)/(3*n+1);

                                       n
                                      x
                     u := n -> -----------------
                               (n + 1) (3 n + 1)

> f:=sum(u(n),n=0..100):
> plot(f,x=-1..1);

> l:=int(1/(1-y^3),y);

                                                       1/2
               2                 1/2        (2 y + 1) 3
  l := 1/6 ln(y  + y + 1) + 1/3 3    arctan(--------------)
                                                  3

         - 1/3 ln(y - 1)

> cste:=simplify(subs(y=0,l));

                                  1/2
                              Pi 3
                      cste := ------- - 1/3 I Pi
                                18

> ll:=simplify((l-cste)/y);

             /                                            1/2
             |      2               1/2        (2 y + 1) 3
  ll := 1/18 |3 ln(y  + y + 1) + 6 3    arctan(--------------)
             \                                       3

                                         \
                             1/2         |
         - 6 ln(y - 1) - Pi 3    + 6 I Pi|/y
                                         /

> h:=subs(y=x^(1/3),ll);

            /
            |      (2/3)    (1/3)
  h := 1/18 |3 ln(x      + x      + 1)
            \

                             (1/3)       1/2
              1/2        (2 x      + 1) 3             (1/3)
         + 6 3    arctan(-------------------) - 6 ln(x      - 1)
                                  3

                           \
               1/2         |   /  1/3
         - Pi 3    + 6 I Pi|  /  x
                           / /

> convert(1/(n+1)/(3*n+1),parfrac);

                            3            1
                       ----------- - ---------
                       2 (3 n + 1)   2 (n + 1)

>