proprietary background service and API package for Android devices from Google
Factsheet
Does Google Play Services for Android update automatically?
What is Google Play Services for Android?
Can I uninstall Google Play Services for Android?
Videos
Recently I reset my ONE M7 and I found that Play Services can't be Updated .Do anyone know how to fix this.. on Play Store it won't show any update button.. I never Saw these Things in my Other 2013 phones like Galaxy S4.
You will need to goto a 3rd party website that hosts Google apps and download the newest (for your device) version. The one I use is http://gappsandroid.blogspot.com/ , you can download EVERY single Official Google app.
You could use OpenGApps. I believe it is sanctioned by Google and allows you to easily select a version for your device architecture and Android version, customizing the package by which apps/services you want.
Once downloaded, you can flash the package via recovery. It is also possible to extract the apps for a manual upgrade; see my answer here for details.