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An explicit bound for the error term of the expansion of Fn(x) := P 1=0 k n x 2 k as x ! 1 i is given. Notation. The symbol -something assumes the value 1 when something holds, 0 otherwise; Bn(¢) is the nth Bernoulli polynomial and Bn is the nth Bernoulli number;bxc,fxg and Imx denote the integer, fractional and imaginary parts of x, respectively; log 2 (x) denotes the base 2 logarithm; at last, the symbol x! 1 i will be used to mean that x2 R, x < 1 and x! 1.

An explicit bound for the error term of the development at s=1 of a set of lacunary series