拉马努金恒等式

1. 3=1+21+31+4…−−−√−−−−−−−−√−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−√

证明就是反复利用平方差公式：

n=1+(n−1)⋅(n+1)−−−−−−−−−−−−−−−−−√

2. 推广到实数域

f(x)=1+x1+(x+1)1+(x+2)1+...−−−−−√−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷

记： fn(x)=1+x1+(x+1)1+(x+2)1+⋯+(x+n−1)1+x+n−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷ ，显然 fn(x) 是关于 n 单调递增的，又：

x+1=1+x(x+2)−−−−−−−−−−√x+1=1+x1+(x+1)(x+3)−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−√…x+1=1+x1+(x+1)1+…+(x+n−1)(1+x+n)−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√

显然 x+1>fn(x) ， fn(x)>1+x1+x1+…−−−−−−√−−−−−−−−−−−√−−−−−−−−−−−−−−−−−√=x+x2+4√2>x 证明见 奇妙的等式 && 精妙的证明（二）；

x<f(x)≤x+1

references

霸气的恒大、霸气的拉马努金恒等式！