What are two examples of divergent sequences?

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Answer 1 · 2015-07-01T23:30:08

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

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.