- #1
Lucretius
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Homework Statement
Determine whether each of the sequences converges or diverges. If it converges, find its limit. Explain with sufficient details each claim.
Homework Equations
(k) [tex]a_n=\sqrt{n^2+1}-n[/tex]
The Squeeze Theorem (If [tex]b_n<a_n<c_n[/tex], and both [tex]\displaystyle\lim_{x\rightarrow\infty}a_n, c_n = L[/tex], then [tex]\displaystyle\lim_{x\rightarrow\infty}b_n=L[/tex])
The Attempt at a Solution
I had the choice of deciding between the squeeze theorem and L'Hopital. L'Hopital seemed useless because I don't see a way to get this into the form of a quotient. I know the sequence is convergent to 0. Figured it would be best to try the Squeeze theorem, as it's really my only other option.
I know that [tex]0 \leq \sqrt{n^2+1}-n[/tex], but after struggling for a while, I still have no idea how to get a larger limit on the other side equal zero (without arbitrarily choosing one, 1/x for instance works, but there's no logical way to get from my function to 1/x...) I know that [tex]n+1>\sqrt{n^2+1}[/tex], but the result is n+1-n=1, and the limit of 1 is 1, period. Likewise, I can get rid of the 1, but I'll have n left over, and the limit of n is infinity! Got any good tips on getting the other side of the squeeze theorem inequality?