一类积分

（1）

\[\sum_{k=1}^n \frac{(-1)^k}{k}\binom{n}{k} =\sum_{k=1}^n(-1)^k\binom{n}{k}\int_0^1 x^{k-1}\text{d}x =-\int_0^1\sum_{k=1}^n \binom{n}{k}(-x)^{k-1}\text{d}x =\int_0^1\frac{1-(1-x)^n}{-x}\text{d}x=-\int_0^1\frac{1-x^n}{1-x} =-\int_0^1\sum_{k=1}^n x^{k-1}\text{d}x=-\sum_{k=1}^n \frac{1}{k}\]

标签：frac,int,积分,sum,text,一类,binom
来源： https://www.cnblogs.com/Math-Nav/p/13450209.html