O15-C903
> restart;
> u:=n->sum(1/k,k=1..n)-ln(n);

> evalf(limit(u(n),n=infinity));

                             .5772156649

> v:=n->cos(n*Pi)/(n+1);

> evalf(sum(v(n),n=0..20));

                             .7163904508

> a:=n->n^2+n+1;

> w:=n->v(2*n)+Sum(v(a(n)+2*'i'),'i'=0..n);

> t:=n->Sum(w('k'),'k'=0..n);

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

      /  0                    \        /  1                    \
      |-----                  |        |-----                  |
      | \    cos((1 + 2 i) Pi)|        | \    cos((3 + 2 i) Pi)|
  1 + |  )   -----------------|, 1/3 + |  )   -----------------|,
      | /         2 + 2 i     |        | /         4 + 2 i     |
      |-----                  |        |-----                  |
      \i = 0                  /        \i = 0                  /

              /  2                    \
              |-----                  |
              | \    cos((7 + 2 i) Pi)|
        1/5 + |  )   -----------------|,
              | /         8 + 2 i     |
              |-----                  |
              \i = 0                  /

              /  3                     \
              |-----                   |
              | \    cos((13 + 2 i) Pi)|
        1/7 + |  )   ------------------|,
              | /         14 + 2 i     |
              |-----                   |
              \i = 0                   /

              /  4                     \
              |-----                   |
              | \    cos((21 + 2 i) Pi)|
        1/9 + |  )   ------------------|,
              | /         22 + 2 i     |
              |-----                   |
              \i = 0                   /

               /  5                     \
               |-----                   |
               | \    cos((31 + 2 i) Pi)|
        1/11 + |  )   ------------------|
               | /         32 + 2 i     |
               |-----                   |
               \i = 0                   /

> for j from 1 to 5 do evalf(t(5^j)) od;

                             .05553132583


                             -.6088787641


                             -1.382402706


                             -2.180753720


                             -2.984193957

> # "La convergence non absolue des séries numériques n'est pas a priori commutative"