In my textbook, it states the following theorem:
If $\lim\limits_{n \to\infty} |a_n|=0$ then $\lim\limits_{n \to\infty} a_n=0$
My question is, if I find that the limit of the absolute value of the sequence does not equal $0$, can I conclude that the limit of the actual sequence then diverges? Or does the theorem only work one way?
If $|a_n|$ does not converge to $0$, then you can certainly say that neither $a_n$ converges to $0$. However, if $|a_n|$ does not converge to $0$, you cannot say that $(a_n)$ diverge. Take for instance as $(a_n)$ the constant sequence equal to 1.
Here are two examples: $$ 2.1,\ 2.01,\ 2.001,\ 2.0001,\ 2.00001, \ \ldots\ldots $$ $$ 2,\ {-2},\ 2,\ {-2},\ 2,\ {-2}, 2,\ {-2},\ 2,\ {-2},\ \ldots $$
In both cases, the absolute value converges to $2$, and $2$ is not $0$.
The first sequence converges.
The second sequence does not converge.
However, if the sequence of absolute values diverges to $+\infty$, then in every case the original sequence (without absolute values) fails to converge.
It is a one way statement. Consider the following sequences:
$$\{+1,-1,+1,-1,\dots\}$$
$$\left\{\frac21,\ \frac32,\ \frac43,\ \frac54,\dots\right\}$$
Then both have $\lim_{n\to\infty}|a_n|=1$, but one diverges while the other converges.
However, notice that if $\lim_{n\to\infty}|a_n|=c$, then the sequence is bounded.
