Interchanging sum and integral (analytic continuation of zeta)

Question

While learning about analytic continuation of $\zeta(s)$, I have come across the following formula $$\pi^{-s/2} \Gamma(s/2)\zeta(s) = \sum_{n=1}^{\infty} \int_{0}^{\infty} e^{-n^2 \pi x} x^{s/2 -1} dx$$ where $\sigma = \text{Re}(s) > 1$. I am trying to use Weierstrass $M$-test to justify the interchanging of the sum and integral but having trouble at $0$. I have come up with the inequality $$\left| \sum_{n = 1}^{\infty} e^{-n^2 \pi x} x^{s/2 - 1} \right| \leq \frac{x^{\sigma/2 - 1}}{e^{\pi x} -1}$$ but this doesn't seem to help me at $0$. Thanks in advance.