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2001 District Olympiad
4
Romania District Olympiad 2001 - Grade XI
Romania District Olympiad 2001 - Grade XI
Source:
March 16, 2011
limit
induction
inequalities
real analysis
real analysis unsolved
Problem Statement
Prove that:a) the sequence
a
n
=
1
n
+
1
+
1
n
+
2
+
…
+
1
n
+
n
,
n
≥
1
a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1
a
n
=
n
+
1
1
+
n
+
2
1
+
…
+
n
+
n
1
,
n
≥
1
is monotonic.b) there is a sequence
(
a
n
)
n
≥
1
∈
{
0
,
1
}
(a_n)_{n\ge 1}\in \{0,1\}
(
a
n
)
n
≥
1
∈
{
0
,
1
}
such that:
lim
n
→
∞
(
a
1
n
+
1
+
a
2
n
+
2
+
…
+
a
n
n
+
n
)
=
1
2
\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}
n
→
∞
lim
(
n
+
1
a
1
+
n
+
2
a
2
+
…
+
n
+
n
a
n
)
=
2
1
Radu Gologan
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