Ok so i am looking for a series convergent for An but noy convergent for Akn (k>1 and integer) for example:

a1+a2+a3+a4+a5+.... is convergent

but a2+a4+a6+a8 .... and a3+a6+a9+a12+.... are not convergent

I know that I need negative terms

I have tried using alternating harmonic series while that works for a2+a4+a6

it doesnt work for a3+a6+a9.. so i need a bit tweaking somewhere there but i dont know what to do/change

also another question i have is:

Does adding 2 convergent series together always produce a convergent one ?