题目

题型：辽宁难度：来源：

lim

n→∞

( 4 5 - 6 7 )+( 4 52 - 6 72 )+…+( 4 5n - 6 7n )

( 5 6 - 4 5 )+( 5 62 - 4 52 )+…+( 5 6n - 4 5n )

=______．

答案

lim

n→∞

( 4 5 - 6 7 )+( 4 52 - 6 72 )+… +( 4 5n - 6 7n )

( 5 6 - 4 5 ) +( 5 62 - 4 52 )+…+( 5 6n - 4 5n )

=

lim

n→∞

( 4 5 + 4 52 +…+ 4 5n )-( 6 7 + 6 72 +…+ 6 7n )

( 5 6 + 5 62 +…+ 5 6n ) -( 4 5 + 4 52 +…+ 4 5n )

=

lim

n→∞

4 5 [1-( 1 5 )n] 1- 1 5 - 6 7 [1-( 1 7 )n] 1- 1 7

5 6 [1-( 1 6 )n] 1- 1 6 - 4 5 [1-( 1 5 )n] 1- 1 5

-
=

lim

n→∞

( 1 5 )n-( 1 7 )n

( 1 6 )n-( 1 5 )n

=

lim

n→∞

1-( 5 7 )n

( 5 6 )n-1

=-1．