On second thoughts, I'm beginning to think that the book may be wrong and I may be right.
You are right and the question is wrong. You can check this by testing the formula with n=2. In that case, the sum becomes
That agrees with your answer (with n=2), but not with the given answer 0.
My guess is that the person who set this question forgot the constant when doing the integration. Congratulations to you for remembering that this is necessary.