Results 1 to 9 of 9

Math Help - monotonic sequence, is it increasing and is it bounded? is my work right?

  1. #1
    Newbie
    Joined
    Sep 2013
    From
    Manorville
    Posts
    5

    monotonic sequence, is it increasing and is it bounded? is my work right?

    an=n+1/n (btw anything that i write with a less than sign i mean less than or equal to i just cant write it.)

    so i figured out that
    a1=2
    a2=2 1/2
    a3=3 1/3

    so my guess is that its increasing.

    so for n=k
    i need to show ak<ak+1
    a1=2<ak=2 1/2 so i showed this right i believe.
    next, i need to show that ak+1<ak+2
    so i did (k+1)+(1/k+1)<(k+2)+(1/k+2)
    cancelled out and reduced to (k+1)+1<(k+2)+1
    reduced to (k+1)<(k+2)
    reduced to 1<2
    is this right to show that it is increasing?
    then i just took the limit of n+1/n and got infinity, so it is not bounded.
    is this all right? my work and all? my professor did it a different way. Thank you guys!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,527
    Thanks
    773

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    Quote Originally Posted by abowler10154 View Post
    so i did (k+1)+(1/k+1)<(k+2)+(1/k+2)
    cancelled out and reduced to (k+1)+1<(k+2)+1
    And what did you cancel out? And please don't say 1/k.

    If a_n=1+1/n, you don't need to use the induction hypothesis, so you don't need to use induction at all. However, I am not sure the definition is not a_n=(n+1)/n.

    In plain text, it is customary to write a_n for a_n and <= for \le.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2013
    From
    Manorville
    Posts
    5

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    I made common denominators on each side, k+1 being the left common one and k+2 being the right one so i could add them individually and cancel if you understand what i mean. But is the way i did it correct also? or no.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2013
    From
    Manorville
    Posts
    5

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    Actually, thinking about it, i dont believe i can cancel...i think im forgetting the rules of 8th grade math in college now. So if i got to the point where i cancelled is that okay? then how would i proceed if so?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,527
    Thanks
    773

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    It seems to me that you omitted parentheses around k+1 in 1/k+1 and treated it as a sum instead of a fraction that it is.

    If n>1, then n+1+\frac{1}{n+1}>n+1>n+1/n. The last inequality holds because 1/n < 1. No use of induction hypothesis, and thus of induction, is necessary.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    Quote Originally Posted by abowler10154 View Post
    an=n+1/n (btw anything that i write with a less than sign i mean less than or equal to i just cant write it.)
    For this kind of sequence think of the function f(x)=x+\tfrac{1}{x}.

    Now f'(x)=1-\tfrac{1}{x^2} which tells us its increasing. But looking it, it is not bounded for x>1.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Sep 2013
    From
    Manorville
    Posts
    5

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    I forgot that i can take the derivitive and if its positive its increasing. Thank you, so now how do i show the work that it is not bounded? Its obviously bounded below as a1 is 2 and its increasing. but what about being bounded above?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    Quote Originally Posted by abowler10154 View Post
    I forgot that i can take the derivitive and if its positive its increasing. Thank you, so now how do i show the work that it is not bounded? Its obviously bounded below as a1 is 2 and its increasing. but what about being bounded above?
    Clearly it is not bounded above.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Sep 2013
    From
    Manorville
    Posts
    5

    Re: monotonic sequence, is it increasing and is it bounded? is my work right?

    ok, another way to show its increasing...how about this?

    an<an+1
    (n^2+1)/n<((n+1)^2+1)/(n+1)
    cross multiply and i get that 2<4. does this prove its increasing?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Family of monotonic strictly increasing functions...
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 24th 2012, 11:13 AM
  2. [SOLVED] Prove that the sequence is increasing and bounded above
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 15th 2012, 12:50 AM
  3. Sequence, Increasing, Not Bounded
    Posted in the Calculus Forum
    Replies: 5
    Last Post: September 24th 2011, 06:44 PM
  4. sequence monotonic? bounded?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 15th 2009, 04:16 AM
  5. bounded increasing sequence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 27th 2008, 03:25 PM

Search Tags


/mathhelpforum @mathhelpforum