Erudition is
the extensive knowledge gained, etc, see previous,
One the other hand,
erudite, the adjective, is describing someone or something that is cultured and sophisticated,
because of their education and background.
You could have erudition, but then you would be described as being erudite.
Being intelligent and persuasive is not erudite. That's just being smart and assertive.
Erudite is persuasive, articulate and suave all mixed in together.
Think Thomas Jefferson, or Marc Anthony from Julius Caesar.