设n∈N*,(2x+1)^n的展开式各项系数之和为an,(3x+1)^n展开式的二项式系数之和为bn,求limn→+∞（2an+3bn）/（an+1*bn+1）的值

Question

设n∈N*,(2x+1)^n的展开式各项系数之和为an,(3x+1)^n展开式的二项式系数之和为bn,求limn→+∞（2an+3bn）/（an+1*bn+1）的值

Answer 1

令x=1由二项式定理可得an=3ⁿ,（3x+1﹚ⁿ展开式的二项式系数之和bn=2ⁿ
∴ limn→∞2an+3bn/an+1bn+1= limn→∞2•3ⁿ+3•2ⁿ/3﹙ⁿ+1﹚+2﹙ⁿ+1﹚= limn→∞2+3•(2／3)ⁿ/3+2•(2／3)ⁿ= 2+3limn→∞(2／3)ⁿ/3+2limn→∞(2／3)ⁿ= 2/3