大家好，欢迎来到IT知识分享网。
1.
3=1+21+31+4…−−−√−−−−−−−−√−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−√
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−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷
fn(x)
n
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
x<f(x)≤x+1
references
霸气的恒大、霸气的拉马努金恒等式！
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