I never get these general term series question, multiplied by some function

I know that simply by the divergence test or that it is the harmonic series.

which is not 0 and by the divergence test if the limit of diverges

Could say that the An is just like multiplying a constant and which multiplying by a constant will not affect if a series diverges or convergences? just the sum?

similarly

This is a Harmonic series which diverges...I am incorrect?