I could not find a Desmos teacher activity exploration for series, so I made my own. Everything is better with Desmos!
This activity works best if students are already familiar with geometric sequences and series. They are really just exploring convergent and divergent. I instruct them to look up the words convergent and divergent in the dictionary. I thought knowing these definitions would help it make sense. I also included a geometric sequence and series link from Math Is Fun at the end.
Desmos Convergent and Divergent Geometric Series Activity.
Enjoy!
I Speak Math 
This is terrific. I wish I’d had this 4 weeks ago when I was teaching this topic. Will definitely use it next year. Thanks for sharing.
I do think you should try them with 0.999 in place of 2 on the desmos page you show. perhaps with 0.5 first !!!
Ok! I will. Thank you for the tip!