若n∈N+,n≥2,求证:12-1n+1<122+132+…+1n2<1-1n.-数学

题目简介

若n∈N+,n≥2,求证:12-1n+1<122+132+…+1n2<1-1n.-数学

题目详情

若n∈N+,n≥2,求证:
1
2
-
1
n+1
1
22
+
1
32
+…+
1
n2
<1-
1
n
题型:解答题难度:中档来源:不详

答案

证明:∵class="stub"1
22
+class="stub"1
32
+…+class="stub"1
n2
> class="stub"1
2×3
+class="stub"1
3×4
+…+class="stub"1
n(n+1)
=class="stub"1
2
 -class="stub"1
3
+class="stub"1
3
-class="stub"1
4
+…+class="stub"1
n
-class="stub"1
n+1
=class="stub"1
2
-class="stub"1
n+1

class="stub"1
22
+class="stub"1
32
+…+class="stub"1
n2
<class="stub"1
1×2
+class="stub"1
2×3
+class="stub"1
3×4
+…+class="stub"1
(n-1)n
=1-class="stub"1
2
+class="stub"1
2
-class="stub"1
3
+class="stub"1
4
-…+class="stub"1
n-1
-class="stub"1
n
<1-class="stub"1
n

所以class="stub"1
2
-class="stub"1
n+1
<class="stub"1
22
+class="stub"1
32
+…+class="stub"1
n2
<1-class="stub"1
n

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