Many convergent sequences imply the initial sequence zero? In connection to this question, I found a similar problem in another Miklos Schweitzer contest:

Problem 8./2007 For $A=\{a_i\}_{i=0}^\infty$ a sequence of real numbers, denote by $SA=\{a_0,a_0+a_1,a_0+a_1+a_2,...\}$ the sequence of partial sums of the series $a_0+a_1+a_2+...$. Does there exist a non-identically zero sequence $A$ such that all the sequences $A,SA,SSA,SSSA,...$ are convergent?

If $SA$ is convergent then $A \to 0$. $SSA$ convergent implies $SA \to 0$.
We have 


*

*$SSA=\{a_0,2a_0+a_1,3a_0+2a_1+a_2,4a_0+3a_1+2a_2+a_3...\}$

*$SSSA=\{a_0,3a_0+a_1,6a_0+3a_1+a_2,10a_0+6a_1+3a_2+a_3...\}$.


I suppose when the number of iteration grows, the coefficients of the sequence grow very large, and I suppose somehow we can get a contradiction if the initial sequence is non-identically zero.

I would suggest you try using the alternating harmonic series.  It is conditionally convergent so you can try rearrangements that might pop out convergent to zero. 
