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You might try to regularize the sum, $$\sum_{n=1}^\infty \frac{ \cos(\alpha n) \sin(n+1) }{n}= -\frac{1}{4} i e^{-i} \left(\ln \left(1-e^{i (\alpha-1)}\right)+\ln \left(1-e^{-i (\alpha+1)}\right)-e^{2 i} \left(\ln \left(1-e^{-i (\alpha-1)}\right)+\ln \left(1-e^{i (\alpha+1)}\right)\right)\right).$$ This is finite for $\alpha\neq 1$.