Results 1 to 4 of 4

Math Help - Flat tail?

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    53

    Flat tail?

    Suppose f:\mathbb{R} \longrightarrow [0,\infty) is continuous and that \int f < \infty. Is it always true that \lim_{x\to\infty}f(x)=0?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by davidmccormick View Post
    Suppose f:\mathbb{R} \longrightarrow [0,\infty) is continuous and that \int f < \infty. Is it always true that \lim_{x\to\infty}f(x)=0?
    Clearly we must have that eventually f(x) is non-increasing. So, let f(x) be non-increasing for x\geqslant N\in\mathbb{N}. Then, f is continuous non-negative non-increasing function. Thus, \int_{N}^{\infty}f<\infty\implies \sum_{n=N}^{\infty}f(n)<\infty and so \lim_{n\to\infty}f(n)=\lim_{x\to\infty}f(x)=0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by Drexel28 View Post
    Clearly we must have that eventually f(x) is non-increasing.
    What do you mean?

    In general, functions f\geq 0 such that \int f<\infty are not non-increasing. Some of them converge to 0, others don't.

    Here's an example. Let me describe the graph of f. It consists of triangular peaks that get thinner and higher : around n, we put a triangular peak of width \frac{1}{n^3} and height n. (Is it clear?)

    The area of the triangle at n is \frac{1}{n^2}. Since \sum_n \frac{1}{n^2}<\infty, the total area under the curve is finite, i.e. \int f(x)dx<\infty. However, f clearly does not converge to 0. We even have f(n)=n\to +\infty for n\in\mathbb{N},n\to\infty.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Laurent View Post
    What do you mean?

    In general, functions f\geq 0 such that \int f<\infty are not non-increasing. Some of them converge to 0, others don't.

    Here's an example. Let me describe the graph of f. It consists of triangular peaks that get thinner and higher : around n, we put a triangular peak of width \frac{1}{n^3} and height n. (Is it clear?)

    The area of the triangle at n is \frac{1}{n^2}. Since \sum_n \frac{1}{n^2}<\infty, the total area under the curve is finite, i.e. \int f(x)dx<\infty. However, f clearly does not converge to 0. We even have f(n)=n\to +\infty for n\in\mathbb{N},n\to\infty.
    I misread the question ...I gave myself extra conditions I didn't have.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. turning 3d objects into flat 2d planes
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: March 12th 2011, 12:02 PM
  2. Volume of outside of flat topped cone
    Posted in the Geometry Forum
    Replies: 2
    Last Post: February 6th 2010, 01:12 PM
  3. Flat
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 29th 2008, 10:20 AM
  4. flat algebra
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 2nd 2008, 01:27 AM
  5. left tail, right tail, two tail
    Posted in the Statistics Forum
    Replies: 2
    Last Post: August 13th 2007, 04:03 PM

Search Tags


/mathhelpforum @mathhelpforum