Results 1 to 3 of 3

Thread: finding the limit.

  1. #1
    Senior Member
    Joined
    Dec 2008
    Posts
    288

    finding the limit.

    How do I prove that the limit of the function

    $\displaystyle limit_{n\rightarrow \infty}(x_1^n+x_2^n+x_3^n+.....+x_k^n)^{(1/n)}$

    where $\displaystyle 0<x_1,x_2,.....,x_k\leq 1$ is equal to $\displaystyle max\{x_1,x_2,.....,x_k\}$

    thanks for any help

    solution

    using the sandwich theorem:

    denoting $\displaystyle x_m=max\{x_1,x_2,.....,x_k\}$ we have

    $\displaystyle \{x_1^n+x_2^n+.....x_k^n\}^{1/n} =x_m\{(\frac{x_1}{x_m}^n+.....+1+....(\frac{x_k}{x _m})^n\}\leq x_m(k)^{(1/n)}$

    and $\displaystyle Limit_{n\rightarrow\infty}x_m(k)^{(1/n)}=x_m$ (this is the upper bound)

    $\displaystyle x_m\{(\frac{x_1}{x_m}^n+.....+1+....(\frac{x_k}{x_ m})^n\}\geq x_m$

    and $\displaystyle Limit_{n\rightarrow\infty}x_m=x_m$

    so by the sandwich theorem we have that $\displaystyle limit_{n\rightarrow \infty}(x_1^n+x_2^n+x_3^n+.....+x_k^n)^(1/n)=x_m$

    Solved this after I posted it (I think)
    Last edited by hmmmm; Oct 23rd 2011 at 01:23 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2010
    From
    Clarksville, ARk
    Posts
    398

    Re: finding the limit.

    Quote Originally Posted by hmmmm View Post
    How do I prove that the limit of the function

    $\displaystyle \lim_{n\rightarrow \infty}(x_1^n+x_2^n+x_3^n+.....+x_k^n)^{(1/n)}$

    where $\displaystyle x_1,x_2,.....,x_k\leq 1$ is equal to $\displaystyle max\{x_1,x_2,.....,x_k\}$

    thanks for any help

    solution

    using the sandwich theorem:

    denoting $\displaystyle x_m=max\{x_1,x_2,.....,x_k\}$ we have

    $\displaystyle \{x_1^n+x_2^n+.....x_k^n\}^{1/n} =x_m\{(\frac{x_1}{x_m}^n+.....+1+....(\frac{x_k}{x _m})^n\}\leq x_m(k)^{(1/n)}$

    and $\displaystyle \lim_{n\rightarrow\infty}x_m(k)^{(1/n)}=x_m$ (this is the upper bound)

    $\displaystyle x_m\{(\frac{x_1}{x_m}^n+.....+1+....(\frac{x_k}{x_ m})^n\}\geq x_m$

    and $\displaystyle \lim_{n\rightarrow\infty}x_m=x_m$

    so by the sandwich theorem we have that $\displaystyle limit_{n\rightarrow \infty}(x_1^n+x_2^n+x_3^n+.....+x_k^n)^(1/n)=x_m$

    Solved this after I posted it (I think)
    Is $\displaystyle 0 < x_1,x_2,.....,x_k$?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Dec 2008
    Posts
    288

    Re: finding the limit.

    yes sorry I just missed that out, will edit it now thanks very much
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding limit of this function, using Limit rules
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Feb 27th 2011, 01:12 PM
  2. Finding the limit
    Posted in the Calculus Forum
    Replies: 9
    Last Post: Jan 23rd 2011, 07:32 PM
  3. Finding a limit. Finding Maclaurin series.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 18th 2010, 10:04 PM
  4. Finding the limit
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Oct 17th 2009, 10:40 AM
  5. help finding the limit..
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 2nd 2008, 06:27 AM

Search Tags


/mathhelpforum @mathhelpforum