设随机变量X服从参数为p的几何分布,试证明:E(1/X)=（-plnp）/(1-p)

X和1/X对应的概率是一样的,都是p*(1-p)^(n-1),那么E(1/X)=∑(1/k)*p*(1-p)^(k-1),其中,k从1到无穷.
E(1/X)=p/(1-p)∑[(1-p)^k]/k
=p/(1-p)∑∫-(1-p)^(k-1)dp
=p/(1-p)∫{-∑(1-p)^(k-1)}dp
=p/(1-p)∫-lim[1-(1-p)^(k-1)]/pdp k趋于无穷,则（1-p)^(k-1）趋于0
=p/(1-p)∫-1/pdp
=-plnp/(1-p)