Serial Killer

How would you catch a serial killer that kills with no motive and murder weapon?

Suppose a crazy psycho and yet intelligent person decides to regularly kill random people. Not just in one city but anywhere in the country. Everytime he totally randomly chooses the victim. Then he goes to spy on that person for a few days, then he waits for the right moment and kills the person with a silenced gun. Or if the target is too hard or risky, he simply goes ahead and randomly chooses someone else.
How would a serial killer like that ever get exposed? I mean unless he makes a stupid mistake by accident, he is basically invincible.

Killers always make mistakes–always.*

HH Holmes: Americas first Serial Killer 1/5