牛顿自然哲学的数学原理的微积分,牛顿第二定律微分方程通解

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−1xsdhmn=1∞∏k=0n−1(α−k)(n−1)!xn     = αxsdhmn=1∞⎛⎝⎜⎜⎜⎜∏k=0n(α−k)n!+∏k=0n−1(α−k)(n−1)!⎞⎠⎟⎟⎟⎟xn     = αxsdhmn=1∞∏k=0n(α−k)(n−1)!(1n+1α−n)xn     = αxsdhmn=1∞∏k=0n(α−k)(n−1)! αn(α−n)xn     = α⎛⎝⎜⎜⎜⎜1xsdhmn=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.

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！