# Increasing/ Decreasing of a Sequence

1. Apr 19, 2012

### trap101

Determine the monotonicity and boundedness of the sequence.

1) 4n/ (4n2 + 1)2

2) 2n/ 4n + 1

Question: I'm having a problem in knowing whether the approach I'm using is providing the right solutions.

in 1) I used the an+1/an and tried to compare their ratios. I end up with: 4n+4/ (4n2 + 8n + 5)1/2 . Now I know this will be less than 1 when I use a few "test values" such as n = 1, 2, etc. But how am I certain that the direction of the sequence won't eventually change?

If I take the derivative of the original sequence I end up with (after simplifying): 16n2 - 2n + 4 = 0. In that equation I can't find any critical points so is it safe to say that the sequence is always increasing based on that logic?

In 2) I did the an+1/an approach and got it was decreasing. But only was I was able to conclude that was from putting in "test values" at the end again. This is when I simplified the ratio to: 2(4n+1)/ 4n+1 +1. I tried to find the derivative and get critical points but I couldn't find anything. What should I do in this case?

Thanks

2. Apr 20, 2012

### Bohrok

Try finding the derivative of 4x/(4x2 + 1)2 again.

3. Apr 20, 2012

### trap101

My mistake, it was actually suppose to be 4x/(4x2 + 1)1/2

4. Apr 20, 2012

### Bohrok

Well find the derivative of that, and what do you get?

5. Apr 21, 2012

### trap101

I get 16n2-2n + 4 = 0

which when i tried to solve for roots i could not find any, even using the -b + (b2- 4ac)1/2/2a formula.

6. Apr 21, 2012

### Bohrok

I'm not sure how you're getting that; try it one more time. Remember that (f/g)' = (gf' - fg')/g2

7. Apr 22, 2012

### trap101

Maybe I should explain how I'm getting that. So after I differentiate I get this:

4(4n2+1)1/2 - 4n(1/2(4n2+1)-1/2) / (4n2+1) = 0

and then I simplify everything and end up with what I got above. Then to solve for possible critical points I used the quadratic formula, which doesn't work.

8. Apr 22, 2012

### trap101

Blimey! I forgot to apply the chain rule to the inside. Ok, but now I ended up with 4 = 0 as my simplified expression. What does that indicate?

9. Apr 22, 2012

### SammyS

Staff Emeritus
That means that the equation has no solution. In this particular case it means that the derivative is not zero anywhere.

Since the function, 4x/(4x2 + 1)(1/2), is differentiable for all real numbers, what does the fact that its derivative is never zero tell you?

10. Apr 22, 2012

### trap101

Well when I was looking over some other questions that I'm having the similar problems with what I noticed is that it indicates that either the sequence is increasing or decreasing through out. To find out which one it does exactly, are you able to just take a couple test values and plug them in to observe the behavior? i.e n= 1, 2, 3, etc. Since you know it has to go in only one direction

11. Apr 22, 2012

### SammyS

Staff Emeritus
That will work, or evaluate the derivative at any one point, since it doesn't change sign.

12. Apr 22, 2012

### trap101

Thanks. One other quick question about types of sequences:

2n/ 4n+1

How do I handle this sort of sequence? I tried the an+1/an approach, but I don't think it's conclusive. When I take the derivative I end up with :

(2n)(4n)(ln 2 - ln 4) = 0

this is after setting the respective powers to eln.

How do I determine behavior from here?

13. Apr 22, 2012

### Dick

You don't have to take any derivatives at all. Just divide numerator and denominator by 2^n. The answer will depend on whether you are asking about 2^n/4^n+1 or 2^n/(4^n+1).

14. Apr 23, 2012

### trap101

I don't fully understand what you mean by dividing by 2n. So if I divide everything by 2n I should get something of this form:

1 / 2n + (1/2n)

Now since this is a sequence I'm only concerned with values of n ≥ 1. With that being the case, the fraction will go to zero eventually......Is that the right interpretation?

15. Apr 23, 2012

### Dick

Yes, it goes to zero. To show it's decreasing you just have to show the denominator is increasing.