No, you shouldn't. Writing is hard. Research is tricky. We are all human and we all are going to make mistakes that end up in print, no matter how hard we try not to. Give other authors grace as a professional courtesy. Do it because it is the kind and humane thing to do. Discovering you've made mistakes in print is extraordinarily painful--even more so when some other author jubulantly declares that you've done so. Better yet, show this kind of grace to other authors because the day will come (oh yes it will, especially if you write for a living) when you need that grace from your fellow authors.
Be kind. It's humane, it's human, and it is entirely countercultural.