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Miklós Schweitzer
1956 Miklós Schweitzer
7
Miklós Schweitzer 1956- Problem 7
Miklós Schweitzer 1956- Problem 7
Source:
October 11, 2015
college contests
Problem Statement
7. Let
(
a
n
)
n
=
0
∞
(a_n)_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
be a sequence of real numbers such that, with some positive number
C
C
C
,
∑
k
=
1
n
k
∣
a
k
∣
<
n
C
\sum_{k=1}^{n}k\mid a_k \mid<n C
∑
k
=
1
n
k
∣
a
k
∣<
n
C
(
n
=
1
,
2
,
…
n=1,2, \dots
n
=
1
,
2
,
…
)Putting
s
n
=
a
0
+
a
1
+
⋯
+
a
n
s_n= a_0 +a_1+\dots+a_n
s
n
=
a
0
+
a
1
+
⋯
+
a
n
, suppose that
lim
n
→
∞
(
s
0
+
s
1
+
⋯
+
s
n
n
+
1
)
=
s
\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s
lim
n
→
∞
(
n
+
1
s
0
+
s
1
+
⋯
+
s
n
)
=
s
exists. Prove that
lim
n
→
∞
(
s
0
2
+
s
1
2
+
⋯
+
s
n
2
n
+
1
)
=
s
2
\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2
lim
n
→
∞
(
n
+
1
s
0
2
+
s
1
2
+
⋯
+
s
n
2
)
=
s
2
(S. 7)
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