Question

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?

Answer 1

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`