O14-914
> restart:
> n:=10:
> a:=n->fsolve(tan(n*x)-1/2/x,x,0..Pi/2/n);

                                                          Pi
        a := n -> fsolve(tan(n x) - 1/2 1/x, x, 0 .. 1/2 ----)
                                                          n

> [seq(a(10^n),n=1..7)];

  [.1313837716, .01540005942, .001567661015, .0001570482230,

                                       -5                -6
        .00001570764911, .1570793185 10  , .1570796013 10  ]

> evalf([seq(10^n*a(10^n),n=1..7)]);

  [1.313837716, 1.540005942, 1.567661015, 1.570482230, 1.570764911,

        1.570793185, 1.570796013]

> evalf([seq((a(10^n)-Pi/2/10^n)*10^(2*n),n=1..7)]);

  [-2.56958611, -3.0790385, -3.135312, -3.14097, -3.1416, -3.142,

        -3.14]

> evalf([seq((a(5^n)-Pi/2/5^n+Pi/5^(2*n))*5^(3*n),n=1..7)]);

       [4.99372647, 6.1194918, 6.26180, 6.2793, 6.29, 6.3, 5.]

> # Il semblerait que a_n admette le DL_3:Pi/2/n-Pi/n^2+2*Pi/n^3+o(1/n^3)
>