Question

Determine whether or not the sequence with the given nth term is monotonic, then discuss its boundedness (i) an=cosn/n ?

Answer 1

First of all `AA,nin N , -1 <= cos n<=1`

Hence , `-1/n <= cosn /n <=1/n`. Since cos n is a periodic function,
the given sequence `{a_n}` decreases and then increases in equal intervals monotonically..

Now as n `-> oo, -1/n and 1/n ->0`
Therefore according to squeeze theorem, the sequence `cos n /n` is convergent and converges to 0.

Since the sequence is convergent, it is bounded. That means it would have a lower bound and a upper bound.