I created Welcome pages for each level as shown at viewtopic.php?f=4&t=1070&start=0 and tested each of these logins with a user that was registered at the appropriate level and the method worked correctly.
I then copied the PayPal button shortcode to a page accessible by free level members and logged in as a user at level 0 and used the PayPal button to upgrade to Level 1.
The PayPal portion of the purchase transaction appeared to have been successfully completed and I was returned to the site when I clicked the "Return to ..." button.
After returning to my website, I closed my browser and reopened it. Then I logged in as the user.
Instead of being taken to the Level 1 welcome page, as I had been when I tested as a level 1 member, I was taken to the Level 0 welcome page.
I waited, and logged in again, only to be taken to the Level 0 welcome page once more.
How do I find my error?
edit: I tried again, this time I used the button code rather than the button shortcode and the system worked. I was upgraded. Still wondering what my errors were in using the shortcode.