Problem 1127 (difficulty: 10/10)

For a sequence \(\displaystyle A=(a_0,a_1,a_2,\ldots)\) of reals let

\(\displaystyle SA=(a_0,a_0+a_1,a_0+a_1+a_2,\ldots) \)

be the sequence of its partial sums \(\displaystyle a_0+a_1+a_2+\ldots\). Can one find a non-zero sequence \(\displaystyle A\) for which the sequences \(\displaystyle A\), \(\displaystyle SA\), \(\displaystyle SSA\), \(\displaystyle SSSA\), ...are all convergent?

Miklós Schweitzer memorial competition, 2007

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