问题 填空题
f(n)=
1
n+1
+
1
n+2
+
1
n+3
+…+
1
2n
,则
lim
n→+∞
n2[f(n+1)-f(n)]
=______.
答案

由题意可得,f(n+1)-f(n)=(

1
n+2
+
1
n+3
+
1
n+4
+…+
1
2n+2
)-(
1
n+1
+
1
n+2
+
1
n+3
+…+
1
2n
)=
1
2n+1
+
1
2n+2
-
1
n+1

lim
n→+∞
n2[f(n+1)-f(n)]=
lim
n→+∞
n2
1
2n+1
+
1
2n+2
-
1
n+1
)=
lim
n→+∞
n2
1
(2n+1)(2n+2)
)=
lim
n→+∞
n2
4n2+6n+2
)=
lim
n→+∞
1
4+
6
n
+
2
n2
)=
1
4+0+0
=
1
4

故答案为

1
4

判断题
单项选择题 A3/A4型题