试题与答案

设函数f(x-1)=x+x2+x3+…+xn(x≠0,1),且f(x)中所有项的

题型:填空题

题目:

设函数f(x-1)=x+x2+x3+…+xn(x≠0,1),且f(x)中所有项的系数和为an,则
lim
n→∞
an
2n
=______.

答案:

函数f(x-1)=x+x2+x3+…+xn=

x(1-xn)
1-x
,所以f(x)=
(x+1)[(x+1)n-1]
x

当x=1时,f(x)中所有项的系数和为an=2×(2n-1)=2×2n-2,

lim
n→∞
an
2n
=
lim
n→∞
2×2n -2
2n
=2-
lim
n→∞
2
2n
=2.

故答案为:2.

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