Question

若

lim

x→1

x2-6x+5

x2-1

=a，则

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)的值为（　　）

A．-2

B．- 1 3

C．- 1 2

D．3

Answer 1

∵a=

lim

x→1

x2-6x+5

x2-1

=

lim

x→1

(x-1)(x-5)

(x-1)(x+1)

=

lim

x→1

x-5

x+1

=-2．
∴

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)=

lim

n→∞

[(-

1

2

)+(-

1

2

)2+(-

1

2

)3+…+(-

1

2

)n]
=

lim

n→∞

(- 1 2 )× [1-(- 1 2 )n]

1+ 1 2

=-

1

3

．
故选B．