Is the $\sum\sin(n)/n$ convergent or divergent? [duplicate]
Solution 1:
The sum of $$\sum_{n=1}^{N} \sin(n) = \frac{\sin(N) - \cot \left( \frac1{2} \right) \cos \left( N \right) + \cot \left( \frac1{2} \right)}{2}$$ which is clearly bounded and hence by generalized alternating series test (also known as Dirichlet's test) the sum converges.
EDIT $$S_N = \sum_{n=1}^{N} \sin(n)$$
$$2\sin \left( \frac1{2} \right) \times S_N = \sum_{n=1}^{N} \left( \cos \left( n- \frac1{2}\right) - \cos \left( n+ \frac1{2}\right)\right) = \cos \left( \frac1{2} \right) - \cos \left( N + \frac1{2} \right)$$
Hence, $$S_N = \frac{\cos \left( \frac1{2} \right) - \cos \left( N + \frac1{2} \right)}{2\sin \left(\frac1{2}\right)}$$