Question

若二項(xiàng)式(x

x

-

1

x

)6的展開式中第5項(xiàng)的值是5，則x=______，此時(shí)

lim

n→∞

(

1

x

+

1

x2

+…+

1

xn

)=______．

Answer 1

(x

x

-

1

x

)6的展開式的通項(xiàng)為Tr+1=

C

r6

(x

x

)6-r(-

1

x

)r=(-1)r

C

r6

x

18-5r

2

令r=4得T₅=C₆⁴x^-1=15x^-1
∴15x^-1=5
∴x=3
∴

lim

n→∞

(

1

x

+

1

x2

+…+

1

xn

)=

lim

n→∞

(

1

3

+

1

32

+…+

1

3n

)=

lim

n→∞

1

2

(1-

1

3n

)=

1

2

故答案為3，

1

2