proprietary background service and API package for Android devices from Google
Factsheet
As a workaround you could sideload the last Google Play Services app.
You can check your version under Settings > Apps > All tab > Google Play Services.
Here in the Google Play Store you will find the latest version number or alternatively here at Google with some more in-depth info.
As a resource you could try and trust (hopefully) androidpolice.com and search for Google Play Services, download the app, allow Unknown sources under settings and sideload (install) the app.
/Update: APKMirror is the 'new' service from androidpolice where you find all apk's. Search here.
But usually after a short time Play Services should update automatically.
Check installed version via terminal:
adb shell
pm dump com.google.android.gms|grep version
Is there a phone# to speak to an actual person? I called 855-466-4438 several times at different times and the message is always "we are currently closed". I called 855-836-3987 and there is no option to speak to someone. I tried online chat but had to answer questions before they would connect me, but I don't have the one piece of info that they need.