1 Find the limit superior and limit inferior of an1n sin npi

1.) Find the limit superior and limit inferior of an=(-1)^n sin n^pi/4.

Solution

|an| = sin npi/4

If n is of the form 4k where k is an integer an =0

When n is even integer and of the form 4k-2, then |an| =1 as |sin pi/2| = |sin3pi/2| =1

Thus inferior of an =-1 and superior of an = +1

and all other values of an lie between these 2 extremes.

Superior an = 1

inferior an = -1