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Theorem
Let α∈R be a real number.
Let x∈R be a real number such that |x|<1 .
Then:
- (1+x)α=∑n=0∞αn−n!xn=∑n=0∞1n!(∏k=0n−1(α−k))xn
where αn− denotes the falling factorial.
That is:
- (1+x)α=1+αx+α(α−1)2!x2+α(α−1)(α−2)3!x3+⋯
Proof
Let R be the radius of convergence of the power series:
- f(x)=∑n=0∞∏k=0n−1(α−k)n!xn
By Radius of Convergence from Limit of Sequence:
- 1R=limn→∞|α(α−1)⋯(α−n)|(n+1)!n!|α(α−1)⋯(α−n+1)|
| 1R | = | limn→∞|α(α−1)⋯(α−n)|(n+1)!n!|α(α−1)⋯(α−n+1)| | |||||||||
| = | limn→∞|α−n|n+1 | ||||||||||
| = | 1 |
Thus for |x|<1 , Power Series Differentiable on Interval of Convergence applies:
- Dxf(x)=∑n=1∞∏k=0n−1(α−k)n!nxn−1
This leads to:
| (1+x)Dxf(x) | = | ∑n=1∞∏k=0n−1(α−k)(n−1)!xn−1+∑n=1∞∏k=0n−1(α−k)(n−1)!xn | |||||||||
| = | α+∑n=1∞⎛⎝⎜⎜⎜⎜∏k=0n(α−k)n!+∏k=0n−1(α−k)(n−1)!⎞⎠⎟⎟⎟⎟xn | ||||||||||
| = | α+∑n=1∞∏k=0n(α−k)(n−1)!(1n+1α−n)xn | ||||||||||
| = | α+∑n=1∞∏k=0n(α−k)(n−1)! αn(α−n)xn | ||||||||||
| = | α⎛⎝⎜⎜⎜⎜1+∑n=1∞∏k=0n−1(α−k)n!xn⎞⎠⎟⎟⎟⎟ | ||||||||||
| = | αf(x) |
Gathering up:
- (1+x)Dxf(x)=αf(x)
Thus:
- Dx(f(x)(1+x)α)=−α(1+x)−α−1f(x)+(1+x)−αDxf(x)=0
So f(x)=c(1+x)α when |x|<1 for some constant c .
But f(0)=1 and hence c=1 .
■
Historical Note
The General Binomial Theorem was announced by Isaac Newton in 1676.
However, he had no real proof.
Euler made an incomplete attempt in 1774, but the full proof had to wait for Gauss to provide it in 1812.
牛顿提出了广义二项式定理，并以此为基础发明了微积分的方法，但对于二项式定理没有给出证明，欧拉尝试过，但失败了，直到1812年高斯利用微分方法得到了证明！
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