Swapping integral and sum using dominated convergence theorem

As mentioned in the comments, what you are trying to do is overkill since monotone convergence is enough. But if you want to use dominated convergence you can do \begin{align} |f_k|&= \sum_{n=1}^k t^{s/2-1}e^{-\pi n^2t}=t^{s/2-1}e^{-\pi t}\,\sum_{n=1}^k e^{-\pi (n^2-1)t}\\ \ \\ &\leq t^{s/2-1}e^{-\pi t}\,\sum_{n=1}^\infty e^{-\pi (n^2-1)t}\\ \ \\ &\leq t^{s/2-1}e^{-\pi t}\,\sum_{n=0}^\infty e^{-\pi nt}\\ \ \\ &=t^{s/2-1}e^{-\pi t}\,\frac1 {1-e^{-\pi t}}\\ \ \\ \end{align}