Abstract. Suppose that $\sum ^\infty_{k = 0} a_k$ is a series with $a_k \ge0$. In a previous paper [Luh] some asymptotic properties were obtained for series of the type $\sum ^\infty_{k = 0} a_k \varphi( \sum ^k_{\nu = 0} a_\nu )$ and $\sum ^\infty_{k = 0} a_k \varphi( \sum ^\infty_{\nu = k} a_\nu )$ where the function $\varphi $ satisfies some natural conditions. It has been first shown by L. Leindler [2] that these asymptotics are best possible if especially the functions $\varphi(t) = t^{- \alpha }$ are considered. In a recent paper the authors have proved the sharpness in the general case too. It is the object of this note to show that the asymptotics obtained in [3] are also best possible with respect to other properties.

AMS Subject Classification (1991): 40A05

Received December 5, 2000, and in revised form February 15, 2001. (Registered under 2833/2009.)