For all we have (why?). Then,
So, for all and is convergent which implies is convergent.
Suppose and is a sequence of complex numbers satisfying . Prove that the two infinite series and are both convergent or both divergent.
First, I'm trying to show that if the first converges, then the second also converges. Setting , I know that all are positive and hence converges. But for , this is still unclear for me.