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Homework Help: Monotonic increasing or monotonic decreasing

  1. Feb 22, 2010 #1
    1. The problem statement, all variables and given/known data
    1. Determine whether the sequence {an} = n+(1/n) is monotonic increasing or monotonic decreasing.


    2. Relevant equations



    3. 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?
     
  2. jcsd
  3. Feb 22, 2010 #2

    Mark44

    Staff: Mentor

    Re: Sequences

    You need to show that an+1 >= an for all n >= some number M. You can't just use the values of a few elements of the sequence.
     
  4. Feb 22, 2010 #3
    Re: Sequences

    What is M? How do I find it?
     
  5. Feb 22, 2010 #4
    Re: Sequences

    Try calculating an+1 - an
     
  6. Feb 22, 2010 #5

    Mark44

    Staff: Mentor

    Re: Sequences

    You get to say what it is.
     
  7. Feb 22, 2010 #6
    Re: Sequences

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

    Since an+1 is greater that an the sequence is monotonic increasing.:confused:
     
  8. Feb 22, 2010 #7

    Mark44

    Staff: Mentor

    Re: Sequences

    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?
     
  9. Feb 22, 2010 #8
    Re: Sequences

    Thank you for all your help. But I am completely lost. But I'll try one more question. Do I solve an+1>= an, for n?
     
  10. Feb 22, 2010 #9

    Mark44

    Staff: Mentor

    Re: Sequences

    Yes.
     
  11. Feb 22, 2010 #10
    Re: Sequences

    Ok, so tons of algebra. Thank you for all your help. :smile:
     
  12. Feb 22, 2010 #11

    Mark44

    Staff: Mentor

    Re: Sequences

    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.
     
  13. Feb 22, 2010 #12
    Re: Sequences

    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
     
  14. Feb 22, 2010 #13

    Mark44

    Staff: Mentor

    Re: Sequences

    There's a mistake in the line above, on the left side.
    How does the line above follow from the line above it?
    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.
     
  15. Feb 22, 2010 #14
    Re: Sequences

    Ok I did an+1-an>=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?
     
  16. Feb 22, 2010 #15

    Mark44

    Staff: Mentor

    Re: Sequences

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

    Leave the first 1 as-is.
     
  17. Feb 22, 2010 #16
    Re: Sequences

    So it equals (2n+1)/(n2+n)>=-1?
     
  18. Feb 22, 2010 #17

    Mark44

    Staff: Mentor

    Re: Sequences

    ***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.
     
  19. Feb 22, 2010 #18
    Re: Sequences

    (n^2+3n+1)/(n(n+1))?
     
  20. Feb 22, 2010 #19

    Mark44

    Staff: Mentor

    Re: Sequences

    DON'T DO ANYTHING WITH THE FIRST 1!!!

    1 + 1/(n + 1) - 1/n = 1 + ?
     
  21. Feb 22, 2010 #20
    Re: Sequences

    1+(1/(n(n+1))? sorry. :cry:
     
  22. Feb 22, 2010 #21

    Mark44

    Staff: Mentor

    Re: Sequences

    You have an incorrect sign. Can you find it? There are only two showing.
     
  23. Feb 22, 2010 #22
    Re: Sequences

    1-(1/(n(n+1)))?
     
  24. Feb 22, 2010 #23

    Mark44

    Staff: Mentor

    Re: Sequences

    YES!!!!

    So here is where we are.
    an + 1 - an
    = 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 an + 1 - an? What does that say about the sequence?
     
  25. Feb 22, 2010 #24
    Re: Sequences

    The sign would be positive.
    The sign would be positive.
    This sequence is monotonic increasing.
     
  26. Feb 22, 2010 #25

    Mark44

    Staff: Mentor

    Re: Sequences

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