Does This Sequence Have a Limit?

[twoflower]

Hi,

suppose this sequence:

$$(-1)^{n} \sqrt{n} \left( \sqrt{n+1} - \sqrt{n} \right)$$

I tried to find the limit and got into this point:

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

According to results, the limit doesn't exist. But how can I find it out? Can it be visible from the point I got to?

Thank you.

 

[matt grime]

take the even terms, they tend to 1. take the odd terms they tend to -1

 

[twoflower]

take the even terms, they tend to 1. take the odd terms they tend to -1

Thank you matt, it's clear now. I see my approach is unnecessarilly complicated..

 

[matt grime]

Whenever you see a (-1)^n always think about n odd and n even to see what happens.

 

[twoflower]

take the even terms, they tend to 1. take the odd terms they tend to -1

Don't they happen to tend to -1/2 or to 1/2, respectively?