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Yes the value $\pi/2$ can be obtained like this.

Let $$ f(x):=\frac {1}{x^2}-\frac{\cot(x)}{x} $$

We may compute $$ \int_0^{\pi/2} f(x)\;dx + \sum_{k=1}^\infty \int_0^{\pi/2}\big(f(\pi+x)+f(\pi-x)\big)\;dx=\frac{\pi}{2} $$ and this converges.