[EricG]

Gmail supports Exchange, but insists on using IMAP. Exchange supports PUSH and more options. Should work with hotmail.com and live.* too.

From settings select Add Account then Exchange.
Enter your user name, but select Manual Setup instead of Next.
Enter your password and change Server to eas.outlook.com.
Account is ready, but I recommend these:
In account settings, change Sync Frequency to Automatic (push).
Enable extra folders in Folder Sync Settings. Change Sync emails from.
Your Name is ignored, set that up from outlook.com web mail.