Calculate the limit n^2/( 1 + 2 + 3 + ... + n ), x->infinity

lim
n^2/(1+2+3+...+n)  n--> inf

We know that 1+2+3+...+n = n(n+1)/2

==> lim 2n^2/n(n+1) = 2lim n^2/ lim n^2+n

 Let us divide by the
highest power n^2

==> 2lim (n^2/n^2) / lim n^2(1+1/n)

Reduce similar:

==> 2 lim(1)/lim (1+1/n)   when n-->
inf

==> 2/1+0 = 2