Limit of this series?

noobishnoob

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How do I find the limit of this series
 

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But what are your thoughts? What have you tried? Please re-read the Read Before Posting thread that's stickied at the top of each subforum and comply with the rules found within. Particularly, please share with us any and all work you've done on this problem, including the parts you know for sure are wrong.

Additionally, please include a few clarifications of the problem statement. You mentioned a limit, but you didn't specify what n is approaching. Is it heading towards \(\displaystyle 0\)? \(\displaystyle 57\pi\)? \(\displaystyle \infty\)? Also, are we dealing with a series or a sequence? That is to say, was the full problem text something like:

Evaluate \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: (-1)^n \dfrac{n}{1+n^2}\)

Or was it something like:

Evaluate \(\displaystyle \displaystyle \lim_{n \to \infty} \: S_n\), where \(\displaystyle S_n = (-1)^n \dfrac{n}{1+n^2}\)

Thank you.
 
But what are your thoughts? What have you tried? Please re-read the Read Before Posting thread that's stickied at the top of each subforum and comply with the rules found within. Particularly, please share with us any and all work you've done on this problem, including the parts you know for sure are wrong.

Additionally, please include a few clarifications of the problem statement. You mentioned a limit, but you didn't specify what n is approaching. Is it heading towards \(\displaystyle 0\)? \(\displaystyle 57\pi\)? \(\displaystyle \infty\)? Also, are we dealing with a series or a sequence? That is to say, was the full problem text something like:

Evaluate \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: (-1)^n \dfrac{n}{1+n^2}\)

Or was it something like:

Evaluate \(\displaystyle \displaystyle \lim_{n \to \infty} \: S_n\), where \(\displaystyle S_n = (-1)^n \dfrac{n}{1+n^2}\)

Thank you.

So for this sequence (not series) limit n approaches infinity.

I tried using standard limits but i'm stuck.
 
So for this sequence (not series) limit n approaches infinity.

I tried using standard limits but i'm stuck.
So the series has the sign oscillating. Forget about that for a moment. What is happening to n/(n2+1) as n goes to infinity. If it goes to real number k\(\displaystyle \neq\)0, then there is no limit as the series will be approaching both k and -k. But what if n/(n2+1) approaches 0 or oo?
 
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