According to The Chicago Manual of Style, persuade is related to actions and often takes the verb “to” before it. If I want to go to a drama movie, but you only like comedies, I have to persuade you to go with me.
Convince is related to ideas. Let's say that you don't know who to support in the Republican primary, so you watch the debates. You come away convinced that none of the contenders represents you. Note that convince is not preceded by the verb “to.”
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