Results 1 to 3 of 3

Math Help - An interesting limit

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    43

    An interesting limit

    \lim \{n!e\}= \:?
    where {.} is the symbol of the fractinal part

    Thank you in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    I think that the important observation is that \displaystyle e=\sum_{k=1}^\infty \frac{1}{k!}, so \displaystyle n!\sum_{k=1}^\infty \frac{1}{k!}=\sum_{k=1}^\infty \frac{n!}{k!} will be very close an integer when n is large. So, \{n!e\}\to 0.

    I haven't thought it all the way through, but I think that you can make it rigorous by looking at the partial sums of the series and using Taylor's theorem. Give it a try.
    Last edited by roninpro; November 22nd 2010 at 11:11 AM. Reason: Corrected notation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by zadir View Post
    \lim \{n!e\}= \:?
    where {.} is the symbol of the fractinal part

    Thank you in advance!
    \displaystyle{e=\sum\limits^\infty_{k=0}\frac{1}{k  !}\Longrightarrow n!e=n!\left(1+1+\frac{1}{2!}+\ldots+\frac{1}{n!}+\  sum\limits^\infty_{k=n+1}\frac{1}{k!}\right)} \Longrightarrow\displaystyle{\{n!e\}=\sum\limits_{  k=n+1}^\infty\frac{n!}{k!}}

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. An interesting limit problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 29th 2010, 08:09 AM
  2. Interesting Limit
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 10th 2009, 10:44 AM
  3. An interesting limit
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 23rd 2008, 11:10 AM
  4. An Interesting Limit...
    Posted in the Calculus Forum
    Replies: 9
    Last Post: May 11th 2008, 02:46 PM
  5. interesting limit
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 29th 2007, 08:58 PM

Search Tags


/mathhelpforum @mathhelpforum