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Miklós Schweitzer
1958 Miklós Schweitzer
6
Miklós Schweitzer 1958- Problem 6
Miklós Schweitzer 1958- Problem 6
Source:
October 23, 2015
college contests
Problem Statement
6. Prove that if
a
n
≥
0
a_n \geq 0
a
n
≥
0
and
1
n
∑
k
=
1
n
a
k
≥
∑
k
=
n
+
1
2
n
a
k
\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k
n
1
∑
k
=
1
n
a
k
≥
∑
k
=
n
+
1
2
n
a
k
(
n
=
1
,
2
,
…
)
(n=1, 2, \dots)
(
n
=
1
,
2
,
…
)
,then
∑
k
=
1
∞
a
k
\sum_{k=1}^{\infty} a_k
∑
k
=
1
∞
a
k
is convergent and its sum is less than
2
e
a
1
2ea_1
2
e
a
1
. (S. 9)
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