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Increasing/ Decreasing of a Sequence

  1. Apr 19, 2012 #1
    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. jcsd
  3. Apr 20, 2012 #2
    Try finding the derivative of 4x/(4x2 + 1)2 again.
     
  4. Apr 20, 2012 #3

    My mistake, it was actually suppose to be 4x/(4x2 + 1)1/2
     
  5. Apr 20, 2012 #4
    Well find the derivative of that, and what do you get?
     
  6. Apr 21, 2012 #5
    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.
     
  7. Apr 21, 2012 #6
    I'm not sure how you're getting that; try it one more time. Remember that (f/g)' = (gf' - fg')/g2
     
  8. Apr 22, 2012 #7
    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.
     
  9. Apr 22, 2012 #8

    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?
     
  10. Apr 22, 2012 #9

    SammyS

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    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?
     
  11. Apr 22, 2012 #10


    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
     
  12. Apr 22, 2012 #11

    SammyS

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    That will work, or evaluate the derivative at any one point, since it doesn't change sign.
     
  13. Apr 22, 2012 #12
    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?
     
  14. Apr 22, 2012 #13

    Dick

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    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).
     
  15. Apr 23, 2012 #14
    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?
     
  16. Apr 23, 2012 #15

    Dick

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