请问∑k*p*[(1-p)^k]看答案是不是=1/p如果是不是就错了k=0→+∞

Question

请问∑k*p*[(1-p)^k]
看答案是不是=1/p
如果是 不是就错了
k=0→+∞

Answer 1

记
S=∑[0,∞] k*p*[(1-p)^k]
=∑[1,∞] k*p*[(1-p)^k]
(因为当k=0时k*p*[(1-p)^k]=0),
则
(1-p)S
=∑[0,∞] k*p*[(1-p)^(k+1)]
=∑[0,∞] [(k+1)-1]*p*[(1-p)^(k+1)]
=∑[0,∞] (k+1)*p*[(1-p)^(k+1)]-∑[0,∞] p*[(1-p)^(k+1)]
(令l=k+1)
=∑[1,∞] l*p*[(1-p)^l]-p*∑[1,∞] (1-p)^l
=S-p*(1-p)/[1-(1-p)]
=S-p*(1-p)/p
=S-(1-p),
即(1-p)S=S-(1-p),所以pS=(1-p),S=(1-p)/p=1/p-1
（你的答案1/p是错的：例如p=1时1/p=1,但求和=0）