Monotonic increasing or monotonic decreasing

[Ki-nana18]

Homework Statement

1. Determine whether the sequence {an} = n+(1/n) is monotonic increasing or monotonic decreasing.

Homework Equations

The Attempt at a Solution

I plugged in some digits and got this
a₁=2
a₂=5/2=2.5
a₃=10/3=3.3333333
a₄=17/4=4.25
a₅=26/5=5.2
I drew the coclusion that it is monotonic increasing. Is that right?

 

[Mark44]

You need to show that a_n+1 >= a_n for all n >= some number M. You can't just use the values of a few elements of the sequence.

 

[Ki-nana18]

n >= some number M

What is M? How do I find it?

 

[VeeEight]

Try calculating a_n+1 - a_n

 

[Mark44]

What is M? How do I find it?

You get to say what it is.

 

[Ki-nana18]

n+1+(1/n+1)>= n+(1/n) @n=3
4.25>3.33333

Since a_n+1 is greater that a_n the sequence is monotonic increasing.

 

[Mark44]

No, that won't do. It's true that n + 1 is always > n (assuming n >0), but 1/(n + 1) < 1/n. If each expression on the left side was larger that the corresponding expression on the right side, then I would buy it.

How do you know that for n = 37, or 503, or whatever, that n + 1 + 1/(n + 1) isn't less than n + 1/n?

 

[Ki-nana18]

Thank you for all your help. But I am completely lost. But I'll try one more question. Do I solve a_n+1>= a_n, for n?

 

[Mark44]

Yes.

 

[Ki-nana18]

Ok, so tons of algebra. Thank you for all your help.

 

[Mark44]

It's hardly "tons of algebra." Unless you think three of so lines constitutes "tons."

Presumably you're in a calculus class if you're asking questions about sequences, so it's reasonble to assume that you have mastered algebra to some extent.

 

[Ki-nana18]

Is this right:
n+1+1/(n+1)≥n+(1/n)
(n+1)^2/(n+1)≥((n^2+1))/n
n(n+1)≥ ((n^2+1))/n (n)
n^2+n≥n^2+1
n≥1

 

[Mark44]

Is this right:
n+1+1/(n+1)≥n+(1/n)
(n+1)^2/(n+1)≥((n^2+1))/n

There's a mistake in the line above, on the left side.

n(n+1)≥ ((n^2+1))/n (n)

How does the line above follow from the line above it?

n^2+n≥n^2+1
n≥1

Another way you can do this is to show that n + 1 + 1/(n + 1) - (n + 1/n) ≥ 0 for all n ≥ M, where you specify what M is.

 

[Ki-nana18]

Ok I did a_n+1-a_n>=0 and I got (n²+3n+1)/(n(n+1))>=0. I'm not sure what to do now. I can't cancel any "n" out b/c its all addition. What do I do now?

 

[Mark44]

n + 1 + 1/(n + 1) - n - 1/n
= 1 + 1/(n + 1) - 1/n
= ?

Leave the first 1 as-is.

 

[Ki-nana18]

So it equals (2n+1)/(n²+n)>=-1?

 

[Mark44]

***Leave the first 1 as-is.***

Just rewrite 1 + 1/(n + 1) - 1/n as an expression it is equal to. I don't want to see any inequality sign yet.

 

[Ki-nana18]

(n^2+3n+1)/(n(n+1))?

 

[Mark44]

DON'T DO ANYTHING WITH THE FIRST 1!

1 + 1/(n + 1) - 1/n = 1 + ?

 

[Ki-nana18]

1+(1/(n(n+1))? sorry.

 

[Mark44]

You have an incorrect sign. Can you find it? There are only two showing.

 

[Ki-nana18]

1-(1/(n(n+1)))?

 

[Mark44]

YES!

So here is where we are.
a_{n + 1} - a_n
= n + 1 + 1/(n + 1) - [n + 1/n]
= 1 + 1/(n+1) - 1/n
= 1 - 1/(n(n + 1))

1/(n(n+1) is at most 1/2, when n = 1. For all other value of n, 1/(n(n + 1)) < 1/2. This is pretty obvious, so probably doesn't need to be proved.

So we're subtracting a positive number that is at most 1/2 from 1. What does that say about the sign of 1 - 1/(n(n + 1))? What does that say about the sign of a_{n + 1} - a_n? What does that say about the sequence?

 

[Ki-nana18]

What does that say about the sign of 1 - 1/(n(n + 1))?

The sign would be positive.

What does that say about the sign of a_{n + 1} - a_n?

The sign would be positive.

What does that say about the sequence?

This sequence is monotonic increasing.

 

[Mark44]

Good. I especially liked it that you were assertive, and didn't add question marks.

 

[Ki-nana18]

Thank you so much for all your help! I do have a tendency of second guessing myself when it comes to math. I will be eternally grateful to you.