I want to ask: Let's say we have the series of non-negative terms $\sum_{n=0}^\infty a_n$ which is convergent. Can I directly say that the series $\sum_{k=0}^\infty a_{2k}$ is convergent because all of its terms contain in the first serie which is convergent and they are non-negative?
And what about if they can be negative?
Yes, it is true, because $0\leqslant\sum_{k=1}^na_{2k}\leqslant\sum_{k=1}^{2n}a_k$.
But if the $a_n$'s can be negative, then it is false. Consider the convergent series$$1-1+\frac12-\frac12+\frac13-\frac13+\cdots$$It converges, but$$-1-\frac12-\frac13-\cdots$$diverges.