Question

an = (1+ sin(1/n))^n Does this sequence converge or diverge?

Answer 1

This sequence converges

Answer 2

$\sin\dfrac1n\le\dfrac1n\implies1+\sin\dfrac1n\le1+\dfrac1n\implies\left(1+\sin\dfrac1n\right)^n\le\left(1+\dfrac1n\right)^n$

You know the right side converges to $e$ as $n\to\infty$, so $a_n$ is bounded from above.

Now, $\sin\dfrac1n$ on its own is a monotonically decreasing sequence approaching 0, which means $1+\sin\dfrac1n$ approaches 1 from above, i.e. $\liminf\left(1+\sin\dfrac1n\right)=1$. For all intents and purposes, you can basically think of $1+\sin\dfrac1n$ as a number larger than 1; call it $M$. For all $M>1$, you have $M^n$ a positive, strictly increasing sequence. It follows, then, that $\left(1+\sin\dfrac1n\right)^n$ must be a strictly increasing sequence.

Therefore $a_n$ must converge to $e$ by the monotone convergence theorem.