Using the definition of convergence, how do you prove that the sequence lim (n + 2)/ (n^2 - 3) = 0 converges from n=1 to infinity?

1 Answer
Dec 22, 2017

You apply the deffinition and then simplify

Explanation:

First of all you need to proof that forallepsilon>0 existsk in NN: forall n >= k , |x_n - 0| < epsilon

So:

|x_n - 0| := |(n+2)/(n^2 - 3) - 0| = |(n+2)/(n^2 - 3)| here you can say that the sequence converges to 0 from n = 1 <=> it converges to 0 from n = 3.
So now you have n >= 3 and |x_n - 0| = |(n+2)/(n^2 - 3)| <= |(n+2)/(n^2 - 4)| = |(n+2)/((n + 2)(n - 2))| = |1/(n - 2)| = 1/(n - 2) < epsilon if you choose any k that verifies 1/(k - 2) < epsilon so k > 1/epsilon + 2