1. Limited time only! Sign up for a free 30min personal 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!

Binomial series

  1. Feb 26, 2015 #1
    The binomial series ##(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + ...## only converges for ##|x| < 1## right?
    Is it true that writing ##(1 + x)^n## differently (i.e. ##x^n (1 + \frac{1}{x})^n##) extends the validity of this series to include values of ##x## such that ##|x| > 1##?
     
  2. jcsd
  3. Feb 26, 2015 #2

    Svein

    User Avatar
    Science Advisor

    No. A binomial series is only defined for a given n, Whatever the value of x, there exists an N such that [itex]nx>1000 [/itex] for n>N.
    No.
     
  4. Feb 27, 2015 #3

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes. Although the series might converge for more values than just ##|x|<1##. Complete details are here: http://en.wikipedia.org/wiki/Binomial_series#Conditions_for_convergence

    Yes, If ##|x|>1##, then your trick can be used. If ##|x|<1##, then your trick doesn't work, but the original series does of course. If ##|x|=1## then the situation is a bit annoying.
     
  5. Feb 27, 2015 #4
    Interesting. I once watched a video in which someone proves that the sum of all natural numbers is ##-\frac{1}{12}##, and in one of the steps, he substituted ##x = 1## into the binomial series. Apparently, anything is possible if you're Euler!
     
  6. Feb 27, 2015 #5

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes, that is valid, but only under conditions. For the standard convergence of series, plugging in ##x=1## is forbidden. But there are other definitions where series do not converge how we they usually do. Under those definitions, you do get ##-1/12##.
     
  7. Feb 27, 2015 #6
    Is it possible to approximate ##\sqrt{2}## using the binomial series?
     
  8. Feb 27, 2015 #7

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes, but not directly. What you can do (for example) is to use ##x=-1/2## to approximate ##1/\sqrt{2} = \frac{\sqrt{2}}{2}##. And then you can easily find ##\sqrt{2}##.

    Edit: I guess you can even do it directly, but convergence won't be very rapid.
     
    Last edited: Feb 27, 2015
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Binomial series
  1. Binomial Series (Replies: 2)

  2. Binomial distribution (Replies: 2)

  3. Verifying Binomials (Replies: 4)

  4. Binomial expansion (Replies: 7)

Loading...