Results 1 to 2 of 2

Math Help - Asymptotic behavior of the number of perfect squares less than x

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    13

    Asymptotic behavior of the number of perfect squares less than x

    Let S_x=|\{ n \le x : \left(\exists m \right) \left( n=m^2\right), n\in \mathbb{N} \}|. Find \mathop {\lim }\limits_{x \to + \infty } \frac{{S_x}}{{x}}.

    I'm thinking that S_x would be the greatest m\in\mathbb{N} such that m \le \sqrt{x} plus 1 or S_x=\left[\sqrt {x} \right]+1.

    The limit would then be \mathop {\lim }\limits_{x \to + \infty } \frac{{S_x}}{{x}}=\mathop {\lim }\limits_{x \to + \infty } \frac{{\left[\sqrt {x} \right]+1}}{{x}}.

    I'm not sure if the argument is valid but if it is, the problem I have is finding \left[\sqrt{x}\right]+1 for cases of x. Any help is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Apr 2009
    From
    Atlanta, GA
    Posts
    409
    Your reasoning is perfectly valid. The percentage of square numbers less than or equal to x is approximately \sqrt(x) in x which tends to zero as x tends to infinity.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Asymptotic behavior of an expression
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 25th 2010, 10:24 AM
  2. Perfect squares of n and n+99
    Posted in the Algebra Forum
    Replies: 9
    Last Post: July 7th 2010, 11:55 AM
  3. perfect squares
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: July 19th 2008, 02:47 PM
  4. asymptotic behavior of integral
    Posted in the Calculus Forum
    Replies: 6
    Last Post: June 18th 2008, 11:47 AM
  5. Perfect Squares
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: March 18th 2007, 06:05 PM

Search Tags


/mathhelpforum @mathhelpforum