Proving a sequence is monotone.

[peripatein]

Hi,

I am unsuccessful at showing that the sequence √n + 1/n is an ascending monotone one, i.e. that a_n+1 > a_n for any n, greater than 2 let's say. I have proven that it is not bounded from above and is bounded from below. Any ideas, suggestions, please?

 

[SammyS]

Hi,

I am unsuccessful at showing that the sequence √n + 1/n is an ascending monotone one, i.e. that a_n+1 > a_n for any n, greater than 2 let's say. I have proven that it is not bounded from above and is bounded from below. Any ideas, suggestions, please?

This appears to be essentially the same as a previous thread you started:

https://www.physicsforums.com/showthread.php?t=649117

 

[peripatein]

Hi,
Except that no one there was able to help me. Perhaps someone could now.

 

[STEMucator]

Consider your sequence as a function.

 

[peripatein]

I am not allowed to. It has to be proven without any reference to functions. Seemingly merely by showing that a_n+1 > a_n for any n greater than 2. Yet I am unable to show that that inequality holds.

 

[STEMucator]

Write out what a_(n+1) actually is.

 

[peripatein]

sqrt(n+1) + 1/(n+1).

 

[STEMucator]

Now what do you do to prove one sequence is larger than the other for all n≥2.

 

[peripatein]

Induction? I have been unsuccessful at proving it via induction as well.

 

[STEMucator]

If a_{n+1} > a_n \Rightarrow a_{n+1} - a_n > 0

 

[peripatein]

And?

 

[STEMucator]

Write out what it means.

 

[peripatein]

sqrt(n+1) + 1/(n+1) - sqrt(n) - 1/n > 0

 

[STEMucator]

sqrt(n+1) + 1/(n+1) - sqrt(n) - 1/n > 0

You could clean it up a bit, but is that sequence greater than zero for every n?

If it helps, roughly around n = 1.2, your sequence geometrically will start to be above the line y=0.

 

[peripatein]

It surely is greater than zero but I am unable to demonstrate it algebrically, alas.

 

[STEMucator]

It surely is greater than zero but I am unable to demonstrate it algebrically, alas.

So, start with n≥2.

For n=2 does the inequality hold?

 

[peripatein]

It does.

 

[Mark44]

This appears to be essentially the same as a previous thread you started:

https://www.physicsforums.com/showthread.php?t=649117

peripatain, do not start a new thread for the same problem. I am locking this thread.