I have a question. How can I check or verify that my geomtry service is working properly?
You can check your ArcGIS Server logs. You can add your server url to the front of this url and paste it into the address bar:
After using the URL I got this
I think this means it is working properly right, because I have no logs refering Geometry service.
Correct, That would mean it is working properly
Retrieving data ...