Applications of the Binomial Theorem

[courtrigrad]

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 $(x+a)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} x^{k}a^{v-k}$. $v$ is a real number. But I guess when $v$ is a positive integer $n$ we get $(x+a)^{n} = \sum_{k=0}^{\infty}\binom{n}{k} x^{k}a^{n-k}$. Why does the series terminate at $n = v$?

Thanks

 

[quasar987]

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

How would you quickly derive the binomial series?

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

$$(1+\epsilon)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} \epsilon^{k}$$

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

$$(x+a)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} x^{k}a^{v-k}$$

of radius of convergence 'a'.

And does the Binomial Theorem follow from the binomial series?

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.

Are there any applications at all of the binomial series/ Binomial Theorem to special relativity?

The binomial series is useful for approximations. When we have an expression of the form $(1+\epsilon)^{v}$ where $|\epsilon |<1$, we can write it as its Taylor series (which is in this case, the binomial series) and make the approximation that $(1+\epsilon)^{v}$ 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 $0<|\epsilon|<1$.

In special relativity, the factor $\gamma$ is of the above form, with $\epsilon =-(v/c)^2$ 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.

I know the binomial series is $(x+a)^{v} = \sum_{k=0}^{\infty}\binom{v}{k} x^{k}a^{v-k}$. $v$ is a real number. But I guess when $v$ is a positive integer $n$ we get $(x+a)^{n} = \sum_{k=0}^{\infty}\binom{n}{k} x^{k}a^{n-k}$. Why does the series terminate at $n = v$?

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

*The definition of $\binom{n}{k}$ used is this: http://en.wikipedia.org/wiki/Binomial_series