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Applications of the Binomial Theorem

  1. Aug 29, 2006 #1
    How would you quickly derive the binomial series? Would you have to use Taylor's Theorem/ Taylor Series? And does the Binomial Theorem follow from the binomial series? Are there any applications at all of the binomial series/ Binomial Theorem to special relativity? I know the binomial series is [itex] (x+a)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} x^{k}a^{v-k} [/itex]. [itex] v [/itex] is a real number. But I guess when [itex] v [/itex] is a positive integer [itex] n [/itex] we get [itex] (x+a)^{n} = \sum_{k=0}^{\infty}\binom{n}{k} x^{k}a^{n-k} [/itex]. Why does the series terminate at [itex] n = v [/itex]?

  2. jcsd
  3. Aug 29, 2006 #2


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    That's a lot of question! Take your breath! :tongue:

    It is not quick and painless but it is simply a result of applying Taylor's expansion theorem to the function of one variable [itex]f(\epsilon)=(1+\epsilon)^v[/itex]. The resulting series is

    [tex] (1+\epsilon)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} \epsilon^{k}[/tex]

    and its radius of convergence is found to be 1. One can then decide to set [itex]\epsilon = x/a[/itex] and multiply both sides of the equation by [itex]a^v[/itex] to get

    [tex] (x+a)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} x^{k}a^{v-k} [/tex]

    of radius of convergence 'a'.

    Yes; in the particular case v=n, a positive integer, the series is finite (the series "terminate" at k=n) and equals the binomial sum. More on that later.

    The binomial series is useful for approximations. When we have an expression of the form [itex](1+\epsilon)^{v}[/itex] where [itex]|\epsilon |<1[/itex], we can write it as its Taylor series (which is in this case, the binomial series) and make the approximation that [itex](1+\epsilon)^{v}[/itex] is equal to only the first few terms of the series. You should be able to see why this is reasonable in the scope of [itex]0<|\epsilon|<1[/itex].

    In special relativity, the factor [itex]\gamma[/itex] is of the above form, with [itex]\epsilon =-(v/c)^2[/itex] and v=-½, so a series expansion is possible and we can make the approximation of keeping only the first few terms in in order to ease our calculations or get insights into the equations.

    Watch what happens to [itex]\binom{n}{k}[/itex] for k>n*. It vanishes. So all terms of the binomial series past k=n are zero.

    *The definition of [itex]\binom{n}{k}[/itex] used is this: http://en.wikipedia.org/wiki/Binomial_series
    Last edited: Aug 29, 2006
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