Question

设Zn=(

1-i

2

)n，n∈Z⁺，记S_n=|Z₂-Z₁|+|Z₃-Z₂|+…+|Z_n+1-Z_n|，则

lim

n→∞

Sn=______．

Answer 1

∵|Z_n+1-Z_n|=|(

1-i

2

)n•

-1-i

2

|=(

2

2

)  n•

2

2

=(

2

2

) n+1，
∴S_n=(

2

2

)2+(

2

2

)3+…+(

2

2

)n+1=

( 2 2 )2[1-( 2 2 )n]

1- 2 2

=

1-( 2 2 ) n

2- 2

．
故

lim

n→∞

Sn=

lim

n→∞

1-( 2 2 )n

2- 2

=

1

2- 2

=1+

2

2

，
故答案为：1+

2

2

．