KöMaL 2007 A441

For an arbitrary sequence of numbers $A=(a_0,a_1,a_2,\ldots)$, let $SA=(a_0,a_0+a_1, a_0+a_1+a_2,\ldots)$ be the sequence of partial sums of the series $a_0+a_1+a_2+\ldots$. Is there any sequence $A$, not constant zero, for which the sequences $A$, $SA$, SSA , SSSA ,... are all convergent?