Each event on a Teamup calendar can be shared easily via the Share menu, both in a browser and in the Teamup app.
When you share an event that is part of a recurring series, you share one event instance only (not the whole event series).
To share an entire recurring event series with someone who is not a user of your Teamup calendar, you could use an iCalendar feed:
An iCalendar feed lets you set up a one-way sync between calendars. With an iCal feed, you can “feed” events from your Teamup calendar to a subscribing calendar automatically.
To set it up:
To make sure they only see the recurring event series, create a new sub-calendar that will be the feed source. Assign the event series to both the original sub-calendar(s), and the newly created feed source sub-calendar.