O14-928
> restart;
> a:=n->int(product(t-k,k=0..n),t=0..1)/(n!);

                              1
                             /      n
                            |    --------'
                            |   '  |  |
                            |      |  |    (t - k) dt
                            |      |  |
                            |      |  |
                           /      k = 0
                             0
                 a := n -> --------------------------
                                       n!

> seq(a(n),n=0..5);

                                   -19        -863
                   1/2, -1/6, 1/8, ---, 3/32, -----
                                   180        10080

> evalf([seq(a(n+1)/a(n),n=0..20)]);# Il semble que R=1 ; preuve :  int(t(1-t),t=0..1)/n < |a_n| < 1

  [-.3333333333, -.7500000000, -.8444444444, -.8881578947,

        -.9132275132, -.9294129007, -.9406891775, -.9489754219,

        -.9553096887, -.9603015198, -.9643318503, -.9676507084,

        -.9704288360, -.9727867100, -.9748117111, -.9765687190,

        -.9781068910, -.9794641351, -.9806701415, -.9817484862,

        -.9827181216]

> l:=sum(a(n)*x^n,n=0..infinity);

           infinity /        1                                \
            -----   |  n    /      (n + 1)                    |
             \      | x    |   (-1)        GAMMA(n + 1 - t)   |
      l :=    )     |----  |   ---------------------------- dt|
             /      | n!   |            GAMMA(-t)             |
            -----   |     /                                   |
            n = 0   \       0                                 /

> h:=int(t*(x+1)^(t-1),t=0..1);# Preuve par thm d'intégration terme à terme

                       -x + ln(x + 1) x + ln(x + 1)
                  h := ----------------------------
                                     2
                            ln(x + 1)  (x + 1)

> evalf(subs(x=-0.7,l));evalf(subs(x=-0.7,h));# Pour vérifier

                             .7791108474


                             .7791108479

> plot(h,x=-1..1);

>