Results 1 to 3 of 3

Thread: show this series defines a continuous function on (0,1]

  1. #1
    Banned
    Joined
    Nov 2008
    Posts
    63

    show this series defines a continuous function on (0,1]

    $\displaystyle \sum_{n=1}^{\infty}{\frac{1}{n^{x+1}}} $
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Master Of Puppets
    pickslides's Avatar
    Joined
    Sep 2008
    From
    Melbourne
    Posts
    5,237
    Thanks
    33
    Quote Originally Posted by silversand View Post
    $\displaystyle \sum_{n=1}^{\infty}{\frac{1}{n^{x+1}}} $
    for it not to be continuous then

    $\displaystyle n^{x+1} = 0 $ on the given interval

    $\displaystyle x+1 = \log_n(0) $ is undefined!

    which is not on $\displaystyle (0,1] $

    therefore must be continuous
    Last edited by pickslides; May 3rd 2009 at 10:07 PM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    6
    The function can be written as...

    $\displaystyle f(x)= \sum_{n=1}^{\infty}\frac{1}{n^{1+x}}= \sum_{n=1}^{\infty} e^{-(1+x)\cdot \ln n}$ (1)

    ... so that for any $\displaystyle x_{0}>0$ it exists an $\displaystyle \epsilon$ with $\displaystyle 0<|\epsilon|<x_{0}$ and for which ...

    $\displaystyle \lim_{\epsilon \rightarrow 0} f(x_{0}+\epsilon)= \lim_{\epsilon \rightarrow 0} \sum_{n=1}^{\infty} e^{-(1+x_{0})\cdot \ln n}\cdot e^{-\epsilon \cdot \ln n}=$

    $\displaystyle = \lim_{\epsilon \rightarrow 0} \sum_{n=1}^{\infty} \frac{1}{n^{1+x_{0}}}\cdot \frac{1}{n^{\epsilon}}= \lim_{\epsilon \rightarrow 0} \sum_{n=1}^{\infty} \frac{1}{n^{1+x_{0}}} = f(x_{0})$ (2)

    Therefore $\displaystyle f(x)$ defined in (1) is a continous function...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show that ||x||= d(x,0) defines a norm on V
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Oct 21st 2011, 09:54 PM
  2. Replies: 0
    Last Post: Feb 26th 2011, 04:03 PM
  3. show that the function is continuous?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 27th 2010, 12:52 PM
  4. Show a function is continuous at every point in Reals
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Oct 20th 2009, 04:45 PM
  5. show this series converge uniformly to a continuous function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 4th 2009, 12:07 AM

Search Tags


/mathhelpforum @mathhelpforum