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.