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What are two examples of divergent sequences?

Answer 1

Chloe Alexander

#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.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.