Results 1 to 2 of 2

Math Help - Show a series converges for any x

  1. #1
    Newbie
    Joined
    Sep 2006
    Posts
    18

    Show a series converges for any x

    Is there a good method for showing that a series converges, if I can't describe the series in general?

    That is, if I know the pattern of the progression, but don't know how to express it for the nth term in terms of elementary functions, is there a surefire way to show it converges?

    Take, for instance, a progression like:

    1 - (1/2)x^2 + (1/2)(1/4)x^4 - (1/2)(1/4)(1/6)x^6 + ...

    I know the pattern, but I can't figure out how to express it in terms of factorials and whatnot.

    Any help is appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,662
    Thanks
    602
    Hello, primasapere!

    1 - \frac{1}{2}x^2 + \frac{1}{2\cdot4}x^4 - \frac{1}{2\cdot4\cdot6}x^6 + \frac{1}{2\cdot4\cdot6\cdot8}x^8 -\hdots

    We can express the denominators.

    For example: . 2\cdot4\cdot6\cdot8 \:=\:(2\cdot1)(2\cdot2)(2\cdot3)(2\cdot4) \:=\:2^4(1\cdot2\cdot3\cdot4) \:=\:2^4\cdot4!

    So the general term is: . (-1)^n\frac{x^n}{2^n\cdot n!} . for n = 0,1,2,3,\hdots

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: October 3rd 2011, 01:12 AM
  2. How do I show the series of an*bn converges?
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 24th 2011, 04:59 PM
  3. Show that sin(a_n) converges
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 7th 2010, 11:16 AM
  4. How do I show that this sum converges?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 19th 2010, 11:55 AM
  5. Show a function series converges
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 13th 2009, 08:40 AM

Search Tags


/mathhelpforum @mathhelpforum