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#