What size beam should be used for a 15' patio cover span?
Late to the party, and I hope your patio turned out well.

For anyone finding this question in the future, the answer could be a 4x10 beam with 14' spacing and a patio span of 15' 2x8 rafters.

https://www.escondido.org/Data/Sites/1/media/pdfs/Building/InfoGuideline8b.pdf

A lot of cities or counties will have standard drawings for simple structures, and you can check building departments for things like live load and dead load requirements (snow and wind). This one seems to be a good example that includes span tables and connection details.