What I was taught in grammar books or lectures is
Don't you want to come?
Does the expression below sound natural to a native speaker?
Do you not want to come?
Is the latter one more informal?
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The use of negative questions is conversational. It implies an understanding or expectation between speaker and listener. It means something like
I had thought you wanted to come, but now you are acting like you don't want to come so please explain yourself.
Given the conversational, and therefore relatively informal nature of negative questions, it is more natural to use the less formal "Don't you want to come?"
Saying "Do you not want to come?" is correct, and means the same, but as it is more formal, it would be used less often.
Answerers are focusing on the more "formal" nature of the second option, but I don't believe that's the key to this.
If I see "Do you not want to come?" written down, I would hear it in my head with an emphasis on "not want", and more than a little incredulity.
"After everything we've said, after all the organisation that we've put in place, after our best-friends-forever pact, and now this! Do you not want to come? Is that it? You really are going to blow us out and not come? You are dead to me!"
They're both correct, but they don't have the same function.
"Don't you want to come?" is a negative question, and the meaning is something like, "I thought you wanted to come, but now I'm not as sure. Can you confirm that I'm right?"
Parent of 6-year-old: Don't you want dessert? You've never skipped dessert in your life!
"Do you not want to come?" is a normal question which happens to have a negative proposition. It literally means what it says, which is a request to know if they don't want to come. Nothing suggests the speaker's previous belief.
Friend at a restaurant: Do you not want dessert? If you're not ordering any, neither will I.