An = (1+ sin(1/n))^n does this sequence converge or diverge?

You know the right side converges to as , so is bounded from above.

Now, on its own is a monotonically decreasing sequence approaching 0, which means approaches 1 from above, i.e. . For all intents and purposes, you can basically think of as a number larger than 1; call it . For all , you have a positive, strictly increasing sequence. It follows, then, that must be a strictly increasing sequence.

Therefore must converge to by the monotone convergence theorem.