> 数学 >
设f(n)=
1
n+1
+
1
n+2
+…+
1
2n
(n∈N),则f(n+1)-f(n)= ___ .
人气:100 ℃ 时间:2020-09-13 12:50:55
解答
∵f(n)=
1
n+1
+
1
n+2
+…+
1
2n
(n∈N),
∴f(n+1)=
1
n+2
+
1
n+3
+…+
1
2n
+
1
2n+1
+
1
2n+2

∴f(n+1)-f(n)=(
1
n+2
+
1
n+3
+…+
1
2n
+
1
2n+1
+
1
2n+2
)-(
1
n+1
+
1
n+2
+…+
1
2n

=
1
2n+1
+
1
2n+2
-
1
n+1

=
1
2n+1
-
1
2n+2

故答案为:
1
2n+1
-
1
2n+2
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