MHB Series Question Concering Convergence/Divergence

[shamieh]

Determine whether the sequence converges or diverges.

$$a_n = \frac{(-1)^{n + 1} n}{n + \sqrt{n}}$$

Ok, so if I'm understanding this correctly, I need to just ignore the sign and think of this particular problem as leading coefficient over leading coefficient to get $$\frac{n}{n} = 1$$?

Also, then wouldn't that mean that $$\lim a_n$$ as $${n\to\infty} = 1$$ ?

So since a limit exists wouldn't it converge to 1? What is the logic that is insinuating that this is divergent? I am not seeing it. Maybe I'm looking at this incorrectly?

 

[alexmahone]

The sequence ends up oscillating between 1 and -1. So, it diverges.

 

[Evgeny.Makarov]

Determine whether the sequence converges or diverges.

The title thread says "Series", but the question says, "sequence". Which is it? Or did you mean, "Serious question"? :)

$$a_n = \frac{(-1)^{n + 1} n}{n + \sqrt{n}}$$

Ok, so if I'm understanding this correctly, I need to just ignore the sign and think of this particular problem as leading coefficient over leading coefficient to get $$\frac{n}{n} = 1$$?

You could do this.

Also, then wouldn't that mean that $$\lim a_n$$ as $${n\to\infty} = 1$$ ?

Here you can't ignore the sign.

So since a limit exists wouldn't it converge to 1?

What do you mean by "it": $a_n$ or $\sum_{k=1}^n a_k$?

 

[shamieh]

Sequence, not Series. Typo.

So in any case where something is oscillating back and forth would $$a_n$$ as $$n$$ approaches $$\infty$$ always be divergent or no?

 

[alexmahone]

Sequence, not Series. Typo.

So in any case where something is oscillating back and forth would $$a_n$$ as $$n$$ approaches $$\infty$$ always be divergent or no?

If a series is oscillating between two (or more) values, it can't converge to one value. So, it is divergent.

 

[shamieh]

Exactly the question I was asking, awesome, thanks.