This question may be a dumb one, but I really want someone to explain this paradox.
I have just seen the popular proof of ‘sum of reciprocal of natural numbers squared equals $\frac{\pi^2}{6}$’. That’s a neat proof, everything’s fine.
The proof used the function $$\frac{\pi cot(\pi z)}{z^2}$$ and integrated it along a infinitely large rectangle centered at the origin. I assume using a circle at origin would do the job as well.
What if I use a circle on the right half of complex plane instead? i.e. $z = R + \frac{1}{2} + Re^{i\theta}$ where $R$ is the radius of the circle.
I have loosely proved integrating the same function along this new contour would also vanish as $R$ approaches infinity. Now, the problem comes: the sum of residues is exactly $1/1^2+1/2^2+1/3^2+...$ but this would equal zero as the integral vanishes(integral = sum of residues multiplied by $2\pi i$).
Could someone explain what I have done wrong?