# Simple proof of the Binomial Theorem for $\mathbb{R}$ [closed]

I have noticed that two of the math books used in high school goes about the binomial theorem in this way:

1. Prove it on integers using induction
2. Generalize it and use it in proofs needed to develop calculus
3. Use calculus to prove the binomial theorem for $\mathbb{R}$

In college level text books (Rudin...) this approach is of cause not taken, but these are often to advanced to show in high school.

There is a rather simple outline here that develops most of the foundation needed for proving the binomial theorem for $\mathbb{R}$. But I was wondering if anyone knows of a simple proof suitable to showing in a high school class for interested although not advanced students ?.

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## closed as not constructive by Qiaochu YuanJun 17 '11 at 10:06

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In my experience, the instances of the binomial theorem used "to develop calculus" all involve integer (in fact, usually positive integer) exponents, so that "circularity" would not enter into it... –  Arturo Magidin Jun 16 '11 at 21:22
What does $x^y$ mean without calculus? –  jspecter Jun 16 '11 at 21:28
As noted, question is not clear. Should be closed unless Lars returns and improves it. –  GEdgar Jun 16 '11 at 21:56
See my answer here. –  Bill Dubuque Jun 16 '11 at 21:58
Despite having taught some version of calculus for too many years, I cannot think of anything in the development that needs to rely on the Binomial Theorem, even for positive integers. For example, the derivative of $x^n$ can be found by factoring $x^n-a^n$, or by induction. (If one uses induction, it is best for weaker classes not to bring out the formal machinery explicitly.) Newton's heuristic discovery of a version of the general Binomial Theorem did play a significant role in his initial development of the calculus, but we need not, and generally do not, imitate him. –  André Nicolas Jun 16 '11 at 22:39

We want to prove the generalized binomial theorem, namely that $$(1+x)^{\alpha}=\sum_{k=0}^{\infty}\binom{\alpha}{k}x^{k}$$where $\binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}.$ Since increasing $\alpha$ by an integer amount is simply multiplying by $(1+x)^{n}$, and is easily dealt with, we need only consider some interval of length $1$. Suppose $\alpha\in[-1,0).$ We can write $$\binom{\alpha}{k}=\frac{(-1)^{k}(-\alpha)(1-\alpha)\cdots(k-1-\alpha)}{k!}.$$Multiplying the top and bottom by $\Gamma\left(-\alpha\right),$and using the properties of Gamma gives $$\binom{\alpha}{k}=\frac{(-1)^{k}\Gamma(-\alpha+k)}{k!\Gamma(-\alpha)}.$$ Recall that for $s>0$, $\Gamma(s)=\int_{0}^{\infty}t^{s-1}e^{-t}dt.$ Then $$\sum_{k=0}^{\infty}\binom{\alpha}{k}x^{k}=\frac{1}{\Gamma(-\alpha)}\sum_{k=0}^{\infty}x^{k}\frac{(-1)^{k}\Gamma(-\alpha+k)}{k!}$$ and by using this definition of $\Gamma(s)$, and switching the order we have
$$\frac{1}{\Gamma(-\alpha)}\int_{0}^{\infty}e^{-t}t^{-\alpha-1}\sum_{k=0}^{\infty}x^{k}\frac{(-1)^{k}t^{k}}{k!}dt.$$Recognizing the series, we have$$\frac{1}{\Gamma(-\alpha)}\int_{0}^{\infty}e^{-t(1+x)}t^{-\alpha-1}dt.$$Substituting $u=t(1+x)$, this becomes $$(1+x)^{\alpha}\frac{\int_{0}^{\infty}e^{-u}u^{-\alpha-1}du}{\Gamma(-\alpha)}=(1+x)^{\alpha},$$which was the desired result.