Convergence of $\zeta(s)$ on $\Re(s)> 1$

Here is another approach. Let $s=\sigma +i \omega$. Then, we can write

$$\begin{align} \sum_{n=1}^\infty \frac{1}{n^s}&=\sum_{n=1}^\infty \frac{e^{-i\omega \log(n)}}{n^\sigma }\\\\ &=\sum_{n=1}^\infty \frac{\cos(\omega \log(n))}{n^\sigma }-i\sum_{n=1}^\infty \frac{\sin(\omega \log(n))}{n^\sigma }\\\\ \end{align}$$

Next, we use the Euler–Maclaurin Summation Formula (EMSF) to write for $\sigma \ne 1$, $\sigma >0$

$$\sum_{n=1}^N \frac{\cos(\omega \log(n))}{n^\sigma }=N^{1-\sigma}\left(\frac{(1-\sigma)\cos(\omega \log(N))}{(1-\sigma)^2+\omega^2}+\frac{\omega \sin(\omega \log(N))}{(1-\sigma)^2+\omega^2}\right)+K_1+O\left(N^{-\sigma}\right)$$

$$\sum_{n=1}^N \frac{\sin(\omega \log(n))}{n^\sigma }=N^{1-\sigma}\left(\frac{(1-\sigma)\sin(\omega \log(N))}{(1-\sigma)^2+\omega^2}-\frac{\omega \cos(\omega \log(N))}{(1-\sigma)^2+\omega^2}\right)+K_2+O\left(N^{-\sigma}\right)$$

for some constants $K_1$ and $K_2$ (that $K_1$ and $K_2$ exist can be shown by analyzing the remainder term in the EMSF). Note, in arriving at the series, we used integration by parts twice in applying the EMSF.

Obviously, we find that for $0<\sigma<1$, the series diverge (for $\omega=0$, only the cosine series diverges), while for $\sigma >1$, the series converge. Finally, it is easy to carry out the analysis for the case $\sigma =1$ to see that the series diverge for $\sigma =1$ (See THIS ANSWER).