Divergence of a modified alternating series I'm currently trying to prove that $\sum_{k=1}^\infty t_k$ diverges, where
$$t_k = \frac{(-1)^k+1}{2k}+ \frac{(-1)^k}{\sqrt{k}}$$
I know that $\sum_{k=1}^\infty \frac{(-1)^k}{\sqrt{k}}$ converges by the alternating series test, and that the series $\sum_{k=1}^\infty \frac{(-1)^k+1}{2k}$ diverges, as it's just a scaled harmonic series interspersed with zeros.
My question is: is it correct to say that the series in question diverges because it is the sum of a convergent series and a divergent series? It seems pretty straightforward if this is allowed, but I'm worried that I'm missing some subtlety about saying $\sum t_k = \sum_{k=1}^\infty \frac{(-1)^k}{\sqrt{k}} +\sum_{k=1}^\infty \frac{(-1)^k+1}{2k}$, (i.e., rearranging the terms in the sum), since the convergent part is only conditionally so.
Would someone please be able to clarify if such a subtlety exists, or if what I've done is alright? Thanks!

If $\displaystyle{\sum_{k=1}^\infty a_k=a, \ \sum_{k=1}^\infty b_k=b}$ then $\displaystyle{\sum_{k=1}^\infty (a_k+b_k)=a+b}$.
Now if your series, $\displaystyle{\sum_{k=1}^\infty t_k}$, converges then the series $\displaystyle{\sum_{k=1}^\infty t_k+s_k}$, where $s_k=\dfrac{(-1)^{k+1}}{\sqrt{k}}$, converges ↯.
