For the following problems determine if the sequences below

For the following problems, determine if the sequences below have limits. In each case, if a limit L exists, given E 0, find N such that for n N one has Irn Ll E. (6) (9) 5m3 72 4 (10) 3n2 1 n2 1

Solution

9) it doesn\'t have limit

since as n -> infinity

x_n goes to infinity

10) limit of x_n = 3

as n goes to infinity

x_n -> 3

since |x_n - 3| = |(3n^2 +1/n^2+1) - 3| = | -2/(n^2+1) | < epsilon

if n^2 + 1 > 2/epsilon

i.e. if n > 2/epsilon

n^2 +1 > 2/epsilon

and x_n - 3 < epsilon

8) x_n = (1+2+....n)/n^2 = n(n+1)/2n^2

as n->infinity

x_n ->1/2

|x_n - 1/2| = |n^2-n -n^2/2n^2| = |-n/2n^2| < epsilon

if n > 2/epsilon