Hi Clive,
Before moving on, I want to confirm the following information with you first.
1. Did you create any other users when using this trial account?
2. Which plan of Office 365 you sign up as a trial account and then purchase it?
As I understand, you have an Office 365 trial account and you are the Administrator, after purchase an Office 365 plan, the trial period ends alerts still appears in the Admin page. Please correct me on time if I misunderstand the issue.
If so, please make sure that you have paid successfully. If you purchase the plan just now, I suggest you logout and login to see if the alert still there.
Based on my experience, when we sign up an Office 365 trial account and then create some users to do some test, all the users we created have been assigned trial licenses at that time. For more detailed information, please refer to the article below: About expired subscriptions. You may decide which users to transfer to paid licenses users.
In additional, if you still have any concerns about this, I suggest you contact our Support team for billing and subscription, you may refer to the link below: http://onlinehelp.microsoft.com/en-us/office365-smallbusinesses/ff652569.aspx
Thanks for your understanding.
Reference
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Buy a subscription after trying Office 365
Thanks,
Eric Sun