Scrape the damaged tape off the entire seam of the drywall with the drywall knife. Cut away any of the tape that doesn't come off easily, using your razor knife. Get the seam completely exposed from end to end.
Press mesh tape along the full length of the seam, affixing it tightly.
Cover the tape with drywall joint compound, spreading it on with your drywall knife. Make the line of compound just wide enough to cover the tape (about four inches). Get it completely smooth and flat.
Allow the compound to dry for about six hours. Spread a second coat of compound over the first, making the line of it slightly wider (6 to 8 inches). Let it dry for about six hours.
Apply a third and final coat of compound, making the seam about 10 inches across, and very thin and flat.
Let it set for six hours. Sand it smooth with a drywall sanding pad. Repaint the ceiling.