In each of the following six exercises first make an educate

In each of the following six exercises, first make an educated guess as to what you think the limit is. Then use the definition of limit to prove that your guess correct. Give an example of a sequence {a_n} which does not converge but for which the sequence {|a_n|} does converge.

Solution

take the sequence {a_n} =<..., -1, 1, -1, 1, -1, ..... > , which clearly does not converge since both -1 and 1 are limits here.

but, {|a_n|} = <...,1,1,1,1,,.....> which clearly converges and limit is 1.