Question

If a sequence is unbounded will it necessarily diverge?

Answer 1

Are you sure the question isn't that "not every bounded series converges"? For example

\+ 1 - 1 + 1 - 1 + 1 ...

Answer 2

The problem is the definition of 'diverge'
Some say that 'diverge' means not having a *finite* limite, so this excludes infinities
Some say that diverge means 'having infinity as its limit'

Answer 3

It depends on the definition of diverge. In some classes it means go to infinity, in some it just means "not converge". Seems like they're thinking the first one. So pick a sequence that's unbounded, but not all the points go that way.

Answer 4

**if a sequence is unbounded it will diverge**, this is **true**.  


the contra positive of this statement is **if a sequence converges it is bounded,** is equivalent and thus also **true**  


the inverse  of this statement, that **if a sequence is bounded it will converge**  is **false** (counterexample: 1,-1,1,-1,1,-1...)

and the converse of this statement that **if a sequence is diverges it is unbounded** is **false**  
(counterexample 1,-1,1,-1,1,-1...)

Answer 5

yes, when talking about sequences in the real numbers, every unbounded sequence diverges.