Is $ \frac{1}{2\sin(n)+0.5\sqrt{n}+1.1}+n$ a Cauchy Sequence?

Question

Is

$a_n= \dfrac{1}{2\sin(n)+0.5\sqrt{n}+1.1}+n$

a Cauchy sequence? If I plot the difference between $a_n$ and $a_{n+1}$ it seems to be shrinking as $n$ goes to infinity, with the difference reaching a limit of 1.

Answer 1

The difference between consecutive terms becomes $1$ as $n \to \infty$, this doesn't mean anything, however. For example, consider $a_{n} = n$ which you know obviously diverges, even though the difference between the terms is $1$. So your sequence is most definitely not Cauchy.

A Cauchy sequence requires $|a_n - a_m| < \epsilon$ for all $m,n > N$ for some integer $N$. It has nothing to do with "consecutive" terms and the difference between terms needs to converge to $0$, not $1$.

Answer 2

It's not enough that the difference reaches a limit of $1$. The difference has to become arbitarily small, but it doesn't.

Indeed, you see that $a_{n+1}-a_n = (n+1)-n + \text{junk} = 1+\text{junk}$, where "junk" approaches $0$ as $n\rightarrow\infty$. The sequence is not Cauchy.