I have a Sony Xperia Z3 compact, and when I trying to broadcast my video live on YouTube from Xperia , I sign in with Google+ account and I received "Error when trying to sign in, please check if Google Play Service is up to date",what to do? ... When I buy my phone I updated all applications.
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.
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.