Find Limit of Sequence: \sqrt{n} - \sqrt{n^2 - 1}

[G01]

I have to find out where this sequence converges or if it converges a all:

$$a_n = \sqrt{n} - \sqrt{n^2 - 1}$$

Now, I can't seem to find a good method to solve this. Would my best bet be to use L'hopitals rule to find the limit of the equivalent function or should I try the squeeze theorem. Thats my question. Thanks for the help.

 

[durt]

It looks like it diverges, so show that it is bounded by a divergent sequence.

 

[G01]

ok how the this sound as a complete solution?

An tends seems to tend toward negative infinity and diverge. I proove this by showing that a function which is always greater than it also tends toward negative infinity. I take the greater function to be the squareroot of n.([tex]f(x) = \sqrt{x}[tex]} This function tends toward negative infinity, so the corressponding sequence, and since this sequence is less than the sequence given, the sequence given will also tend toward negative infinity.

Of course in my solution i'll use more mathmatical notation, but is that the correct reasoning?

 

[durt]

Yeah that's right. Except I think you mean the negative square root: $$f(x)=-\sqrt{x}$$. And make sure you have in there: $$f(n) > a_n$$ for all $$n > a$$ (where you find a).