Convergence/Divergence and Monotonicity of a Sequence

[B18]

Homework Statement

Show the following sequence to diverge, or converge. Determine if monotonic.
a sub n=n+(1/n)

The Attempt at a Solution

I understand that the sequence does diverge. I found this because the limit as n→∞ the limit is going to ∞ as well.
I found that the sequence is monotonic by showing that a sub n is less than a sub n+1. I tried to do the first derivative test for this and was confused when I had the sequence decreasing from 0 to 1 and increasing from 1 to ∞. Wouldn't this make the sequence not monotonic? Just need an explanation to this.

 

[LCKurtz]

Homework Statement

Show the following sequence to diverge, or converge. Determine if monotonic.
a sub n=n+(1/n)

The Attempt at a Solution

I understand that the sequence does diverge. I found this because the limit as n→∞ the limit is going to ∞ as well.
I found that the sequence is monotonic by showing that a sub n is less than a sub n+1. I tried to do the first derivative test for this and was confused when I had the sequence decreasing from 0 to 1 and increasing from 1 to ∞. Wouldn't this make the sequence not monotonic? Just need an explanation to this.

Surely you could show $a_n<a_{n+1}$ without resorting to calculus. But the sequence only has values on the integers 1,2,... so it would still be increasing.

 

[B18]

Surely you could show $a_n<a_{n+1}$ without resorting to calculus. But the sequence only has values on the integers 1,2,... so it would still be increasing.

So we are saying since sequences are only accountable for positive integers meaning the negative result on the first derivative test from 0 to 1 has no effect on monotocity.

 

[Jorriss]

Why did you try the first derivative test on a sequence??

 

[B18]

Why did you try the first derivative test on a sequence??

To determine if the sequence is increasing or decreasing to show it is monotonic

 

[LCKurtz]

So we are saying since sequences are only accountable for positive integers meaning the negative result on the first derivative test from 0 to 1 has no effect on monotocity.

Yes. And I'm also saying that using the derivative test on $x+\frac 1 x$ is way overkill, not to mention the fact that it confused the issue for you.

 

[B18]

Would I be correct in saying this sequence has a lower bound of 0 but no upper bound?

 

[LCKurtz]

Would I be correct in saying this sequence has a lower bound of 0

Is $a_n>0$ for all n? What do you think? Is that the greatest lower bound?

but no upper bound?

Is $a_n > n$? Again, what do you think?

 

[B18]

Yes I believe that 0 is the greatest lower bound. And after plotting a couple of points on a graph of this sequence it appears this is no least upper bound.

 

[LCKurtz]

Yes I believe that 0 is the greatest lower bound. And after plotting a couple of points on a graph of this sequence it appears this is no least upper bound.

The problem with "beliefs" is that they are sometimes wrong. And saying "it appears" is a long way from a proof. If this is a homework problem, you have a ways to go before handing it in.

 

[B18]

Well because there is not an m that is ≥ a sub n this tells me there is no least upper bound. I know the greatest lower bound is 0 because the sequence never goes below 0 and therefore there is an M that is ≤ to all a sub n. according to my class notes this is as much evidence as I can provide. That we have learned thus far.

 

[LCKurtz]

Well because there is not an m that is ≥ a sub n this tells me there is no least upper bound. I know the greatest lower bound is 0 because the sequence never goes below 0 and therefore there is an M that is ≤ to all a sub n. according to my class notes this is as much evidence as I can provide. That we have learned thus far.

That argument shows 0 is a lower bound, but not that it is the greatest lower bound.

 

[B18]

Do you have a quick explanation as to what makes a bound the least or greatest? Not seeing how 0 isn't the greatest lower bound when the sequence cannot go below that.

 

[LCKurtz]

Do you have a quick explanation as to what makes a bound the least or greatest? Not seeing how 0 isn't the greatest lower bound when the sequence cannot go below that.

What is the definition for x to be the greatest lower bound of a set S of numbers?