f[x_] := 1 / (1 + 25 * x^2)

data =
Table[
Table[
{1/2^i, f[1/2^ i]},
{i, 1, n}
],
{n, 1, 10}
]

data[[1]]

int[i_, h_] := InterpolatingPolynomial[data[[i]], h] // N

points = Table[
{i, int[i, 0]},
{i, 1, 10}
]

ListPlot[points]