The sequence ak must be convergent if ak is real all k great

The sequence {a_k} must be convergent if: {a_k} is real all k greaterthanorequalto 1. {a_k} is monotone increasing and a_k > 0 for all k greaterthanorequalto 1. {a_k} is bounded. {a_k} is bounded above and monotone increasing. |a_k| lessthanorequalto 1.

Solution

a)

Let, a_k=k

This is a divergent sequence

b)

a_k=k, this is monotone increasing and divergent

c)

a_k=(-1)^k

This is bounded by does not converge

d)

By Monotone Convergence theorem a sequence bounded above and monotone increasing is convergent

e)

a_k=(-1)^k

|a_k|=1

But it does not converge