ABSTRACT. We show that there exists a function in a ball algebra such that almost every slice function has a series of Taylor coefficients divergent with every power p < 2.In §7.2 of [3] W. Rudin gives some examples of boundary behavior of holomorphic functions in complex balls of dimension 2 and 3. It was observed in [4, Remark 1.10], that using Theorem 1.2 of [4] such examples can be constructed in arbitrary dimension. In the present note we further pursue this idea. It is well known that in one variable there exists a function f(z) = Yln°=o anzn analytic for \z\ < 1 such that £ |an|p = oo for all p < 2. Our theorem generalizes this fact to functions of several variables.In this note, B will always denote the unit ball in the complex d-dimensional space Cd, S will stand for the unit sphere and a for the rotation invariant probability measure on S. A(B) will denote the ball algebra of all functions analytic in B and continuous in B. For 0 < p < oo, ||/||p denotes (/s |/(c)|pder(<))l/p. If / is a holomorphic function in B then it has a unique homogeneous expansion as / = 2^£Lo fnt fn i8 an analytic polynomial homogeneous of degree n.It was shown in [4, Theorem 1.2], that there exist polynomials (pn) homogeneous of degree nonfl such that (*) l|Pr,|| = 1 and ||Pn||oo