Does the series ∑n=1∞sin(n)n\sum_{n=1}^{\infty} \frac{\sin(n)}{n}∑n=1∞nsin(n) converge? Identify the most direct test.
Ratio Test shows L=0L = 0L=0; series converges
Direct Comparison Test with 1n2\frac{1}{n^2}n21 proves convergence
Dirichlet's Test: 1n\frac{1}{n}n1 decreases to 0 and ∑sin(n)\sum \sin(n)∑sin(n) has bounded partial sums; series converges
Divergence Test shows limn→∞sin(n)n=0\lim_{n \to \infty} \frac{\sin(n)}{n} = 0limn→∞nsin(n)=0; diverges