求∑(∞,n=1)n(n+1)/2^n的和.

Question

求∑(∞,n=1)n(n+1)/2^n 的和.

Answer 1

考虑幂级数∑(∞,n=1)n(n+1)x^n,
它的收敛区间为(-1,1),
设其和函数为S(x),
则S(x)=∑(∞,n=1)n(n+1)x^n
=x(∑(∞,n=1)x^(n+1))′
=x(x²/(1-x))″
=2x/(1-x)³.
故
∑(∞,n=1) n(n+1)/2^n
=S(1/2)
=(2*1/2)/(1-1/2)³
=8.