Question

级数(2^n+3^n)/6^n等于多少？

n=1到无穷

Answer 1

解答 = ∑n=1 -> n = 无限 [(2/6)^n + (3/6)^n]
又等於(2/6)/(1-2/6) + (3/6)/(1-3/6) = 1/2 + 1 = 3/2
另：
等比级数啊
|q|<1,那麼当n→∞时Sn=a1/(1-q)
这里化简成了求级数1/2^n+1/3^n,和的极限等於极限的和,1/2+1/4+1/8+...=1/2/(1-1/2)=1,1/3+1/9+1/27+...=1/3/(1-1/3)=1/2,所以S=3/2

Answer 2

lim(n->∞) ∑(i:1->n) (2^i+3^i)/6^i
=lim(n->∞) ∑(i:1->n) [(1/3)^i+(1/2)^i ]
=lim(n->∞) [(1/3)( 1- 1/3^n)/(1-1/3) +(1/2)( 1-1/2^n)/(1-1/2) ]
=lim(n->∞) [(1/2)( 1- 1/3^n) +( 1-1/2^n) ]
=1/2 +1
=3/2