two hot wires into a capacitor
Don’t start changing the wires! 
Those pens are not a good way to troubleshoot.
Don’t start shotgunning parts without some basic troubleshooting.
For example I have a 3 speed fan that worked great all summer. One cool day we turned it off, another day it would not turn on at any of the speeds. I unplugged it opened it and spun the blades , they were stiff, it did not turn a full 360. I dribbled some light oil in the bearing close to the blades while turning by hand it freed up. I continued very slowly adding oil and spinning , wiped the excess oil and put it back together. The fan still works every day my grandson turns it on after school to see if it still works. 

In my experience small fan motors fail because of dry bearings almost never the switch or capacitor. I have brought dozens if not hundreds of fans to life using light oil (will get slammed but wd40 works to get thing moving) 3 in 1 is better but a simple 30 weight oil will also work and last longer but takes a little get to get things free.