Results 1 to 5 of 5

Math Help - Calculus: Art Of Problem Solving (2), (3), (4)

  1. #1
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Calculus: Art Of Problem Solving (2), (3), (4)

    Hint for Problem 1: the answer is 2n. Here are 3 more problems:


    \star \star \star Problem 2: Evaluate \lim_{n\to\infty} \left[\prod_{k=1}^n \binom{n+k}{n}\right]^{n^{-2}}.

    \star \star Problem 3: Evaluate \lim_{n\to\infty} \frac{1}{n^k} \int_1^b \prod_{j=1}^k \ln(1 + a_jx^n) \ dx, where k, \ a_j > 0, \ b > 1 are given constants.


    \star \star Problem 4: For any real number x>0 define f(x) to be the number of elements of the set \{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \ m^2 + n^2 < x \}.

    Find \lim_{x\to\infty}\frac{f(x)}{x}.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by NonCommAlg View Post


    \star \star Problem 4: For any real number x>0 define f(x) to be the number of elements of the set \{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \ m^2 + n^2 < x \}.

    Find \lim_{x\to\infty}\frac{f(x)}{x}.
    This hardly needs a thought:

    \lim_{x\to\infty}\frac{f(x)}{x}=\pi

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by CaptainBlack View Post
    This hardly needs a thought:

    \lim_{x\to\infty}\frac{f(x)}{x}=\pi

    RonL
    the answer is correct but the problem is not trivial at all!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jun 2008
    Posts
    170
    Which book are you using? Is the the AoPs volume 1 and volume 2?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by NonCommAlg View Post
    the answer is correct but the problem is not trivial at all!
    But it is, it is equivalent to the definition of area as the limit of the sum of the area of the squares of a regular rectangular partition of the circle.

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: January 6th 2011, 01:27 AM
  2. Modelling and Problem Solving Calculus
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 17th 2010, 06:02 PM
  3. Calculus: solving an equation
    Posted in the Math Challenge Problems Forum
    Replies: 5
    Last Post: May 21st 2009, 06:28 AM
  4. Solving College Calculus 1 Problems?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 5th 2009, 06:12 PM
  5. Calculus: Art Of Problem Solving (1)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 21st 2008, 09:16 PM

Search Tags


/mathhelpforum @mathhelpforum