1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Convergence of a series

  1. May 17, 2006 #1
    Dear all

    If a series e.g. a power series results in x convergering towards zero, can then one conclude that this series converge for all number if lets x belongs to R?

    Sincerely Yours
    Hummingbird25
     
  2. jcsd
  3. May 17, 2006 #2
    what do you mean by R? the ratio between terms?
     
  4. May 17, 2006 #3
    I believe Hummingbird was referring to the set of real numbers (often denoted by R). The answer to the question (if I read it correctly, I had to read it a few times) is no also, look up "radius of convergence".
     
  5. May 17, 2006 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    This makes no sense at all. "A power series results in x converging towards zero"? First of all, x does not "converge" toward anything. It is a variable. Second, I don't know what you mean by saying "a power series results" in that.

    If I really had to guess, I would guess you are asking about the "ratio test". If, for any series of positive numbers
    [tex]\Sum_{n=0}^\infnty a_n[/tex] the sequence [tex]\frac{a_{n+1}}{a_n}[/tex] converges to any number less than 1, then the series converges.

    From that it follows that if, for the power series [tex]\Sum_{n=0}^\infty a_nx^n[/itex] and some specific x, the ratio [tex]\left|\frac{a_{n+1}}{a_n}\right|\left|x\right|[/tex] is less than 1 then the series converges for that x. In particular, if [tex]\left|\frac{a_{n+1}}{a_n}\right|[/tex] converges to 0 then the above will converge to 0 <1 for all x and so the power series converges for all x.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Convergence of a series
  1. Series Convergence (Replies: 3)

  2. Convergence of series (Replies: 2)

  3. Series convergence (Replies: 26)

  4. Convergence of a series (Replies: 24)

  5. Series Convergence (Replies: 15)

Loading...