Do any of these sequences converge? I have a homework question asking which of these 5 sequences converges, but it seems to me that none of them actually converge.
The 5 sequences are:
A) $a_n = n + \frac3n$
B) $\displaystyle a_n = -1 + \frac {(-1)^n} n$
C) $\displaystyle a_n = \sin \frac {n\pi}2$
D) $\displaystyle a_n = \frac {n!} {3^n}$
E) $\displaystyle a_n = \frac n {\ln(n)}$
As far as I can see:


*

*A fails the divergence test as there is no limit

*B also fails the divergence test as the limit is -1

*C would oscillate forever because nothing is limiting the sin function

*D I used the ratio test and ended up with a limit that had no limit so it diverges

*E also fails the divergence test with no limit
Am I missing something or is there a mistake with this question?
 A: You should not be using divergence test for determining whether a sequence converges. Divergence test is a criterion used for determining whether a series converges. (It says that necessary condition for $\sum_{n=1}^{\infty} a_n$ to converge is to have $a_n\to 0$ as $n\to\infty$).
In this case, out of the choices, only B) converges as a sequence. The others diverges, but not for the reasons of divergence test. For example:
A) $a_n$ diverges because $a_n > n$ for each $n\in\mathbb{N}$, and so it is unbounded, so it cannot converge. (Recall that convergent sequence must always be bounded)
B) converges because as you said $a_n\to -1$ as $n\to\infty$.
C) converges, and your reason for this is correct, but it needs to be made precise. (See jiboune's comment, which is using the following fact: in order to show that a sequence is not convergent, it is enough to find two different subsequences converging to different limits. This is justified because if the sequence were convergent, all the subsequences would converge to the same limit).
D) and E) also don't converge roughly for the same reasons as A). Namely, the terms of the sequence grows unbounded. I will leave the necessary verifications to you.
