Question

What are two examples of divergent sequences?

Answer 1

`U_n = n` and `V_n = (-1)^n`

Explanation:

Any series that is not convergent is said to be divergent

`U_n = n` :

`(U_n)_(n in NN)` diverges because it increases, and it doesn't admit a maximum :

`lim_(n->+oo) U_n = +oo`

`V_n = (-1)^n` :

This sequence diverges whereas the sequence is bounded :
`-1 <= V_n <= 1`

Why ?

A sequence converges if it has a limit, single !

And `V_n` can be decompose in 2 sub-sequences :

`V_(2n) = (-1)^(2n) = 1` and
`V_(2n+1) = (-1)^(2n+1) = 1 * (-1) = -1`

Then : `lim_(n->+oo) V_(2n) = 1`
`lim_(n->+oo) V_(2n+1) = -1`

A sequence converges if and only if every sub-sequences converges to the same limit.

But `lim_(n->+oo) V_(2n) != lim_(n->+oo) V_(2n+1)`

Therefore `V_n` doesn't have a limit and so, diverges.