which APK for Google play services? On my Sony Xperia Z1 Compact running CyanogenMod, I had the Google Play Services installed as an APK. At the time I must have thought that this was better than installing GApps - for being able to pick packages myself.

Since a recent update (of either Google play services or CyanogenMod) I get the error "unfortunately Google play services have stopped working". 

Google play store also requires these services because it relies on these services. In the end I removed both the Services and the Store.

I have tried installing the a number of versions of the Services (from 7.3.xx to 8.3.xx) but they also report the same error message after a while.

Does the app need some sort of reset before it will work again? Each time an APK is uninstalled then its cache etc must be removed? 

Or should I install a very specific version of the APK? my build is cm_amami_userdebug 5.1.1 LMY49J d8f66af955 

EDIT:

For flashing GApps, I would choose Open GApps pico for Android 5.1 on ARM(32), does that sound right? 

EDIT 2:

Flashing the GApps above worked. I would still like to know the answer to the first question - whether an APK could have solved it (and which)
The APKs for Google Play Services can be found on APK Mirror, along with explanations of the build numbers and how to determine the right build to install.