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No you do not NEED a Dinner Jacket. I like to wear one, but just a suit and tie would do. Since this ship is of an American line and cruising out of Copenhagen, it will be full of mainly American cruisers, and I expect that they will treat the formal evenings with less enthusiasm than most British people do. Don't splash out on a DJ unless you want to make a point about dressing up for formal nights, as I most certainly would! Happy cruising.
I always wear a plain dark suit with an inexpensive white formal shirt (anything with the buttons concealed) and impressive bow tie (try M&S, BHS for both or Tie Rack for the bow tie). The suit will do for work or general wear when you return rather than just sit in the wardrobe and gather dust. The suit will also do for the semi formal evenings with a different colour shirt and tie. Don't forget you have a weight limit on the flight so every saving helps. Have a nice holiday.