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Discuss the convergence or divergence of the series \sum _{{n=0}}^{{\infty }}\sin n.

The sine function is a periodic function with period length of 2\pi . So, as n approaches \infty , the terms of this series just move around inside the interval [-1,1]. They don’t go anywhere. For example,

\sin 1\approx 0.8415

\sin 2\approx 0.9093

\sin 3\approx 0.1411

\sin 4\approx -0.7568

\sin 100\approx -0.5064

\sin 1000\approx 0.8269

\sin 10000\approx -0.3056

\sin 100000\approx 0.0358

Since the terms do not converge to 0 as n\rightarrow \infty , this series diverges by the nth term test.

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