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