I’ve recently started learning about sequences and convergence and divergence, and I came across the comparison test for sequences. What I have is that:

[![enter image description here](https://i.stack.imgur.com/skt4k.png)](https://i.stack.imgur.com/skt4k.png)

What if $a\_n$ is defined as a periodic function, such as $a\_n = 1 + sin (x)$

If $b\_n$ is any function that is greater than $a\_n$ that does converge, shouldn't $a\_n$ converge?

As an example, let $b\_n = (3x+2)/(x+1)$

This seems to satisfy the conditions, yet $a\_n$ (the sinusoidal function) does not appear convergent...

[![enter image description here](https://i.stack.imgur.com/FYlKJ.png)](https://i.stack.imgur.com/FYlKJ.png)