Question

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

Answer 1

`\sin\frac1n\le\frac1n\implies1+\sin\frac1n\le1+\frac1n\implies\left(1+\sin\frac1n\right)^n\le\left(1+\frac1n\right)^n`

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

Now,
`\sin\frac1n` on its own is a monotonically decreasing sequence approaching 0, which means
`1+\sin\frac1n` approaches 1 from above, i.e.
`\liminf\left(1+\sin\frac1n\right)=1`. For all intents and purposes, you can basically think of
`1+\sin\frac1n` 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\frac1n\right)^n` must be a strictly increasing sequence.

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

Answer 2

This sequence converges