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Miklós Schweitzer
1958 Miklós Schweitzer
11
Miklós Schweitzer 1958- Problem 11
Miklós Schweitzer 1958- Problem 11
Source:
October 23, 2015
college contests
Problem Statement
11. Let
a
n
=
(
−
1
)
n
(
n
=
1
,
2
,
…
,
2
N
)
a_n = (-1)^n (n= 1, 2, \dots , 2N)
a
n
=
(
−
1
)
n
(
n
=
1
,
2
,
…
,
2
N
)
. Denote by
A
N
(
x
)
A_{N}(x)
A
N
(
x
)
the number of the sequences
1
≤
i
1
<
i
2
<
⋯
<
i
N
≤
2
N
1 \leq i_1 < i_2< \dots <i_N \leq 2N
1
≤
i
1
<
i
2
<
⋯
<
i
N
≤
2
N
such that
a
i
1
+
a
i
2
+
⋯
+
a
i
N
<
x
N
2
(
−
∞
<
x
<
∞
)
a_{i_1}+a_{i_2}+ \dots +a_{i_N}< x \sqrt{\frac{N}{2}} (-\infty < x < \infty)
a
i
1
+
a
i
2
+
⋯
+
a
i
N
<
x
2
N
(
−
∞
<
x
<
∞
)
. Show that
lim
N
→
∞
A
N
(
x
)
(
2
N
N
)
=
1
2
π
∫
−
∞
∞
e
−
u
2
2
d
u
\lim_{N \to \infty} \frac{A_{N}(x)}{\binom{2N}{N}} = \frac {1}{\sqrt {2\pi}} \int_{-\infty}^{\infty} e^{-\frac{u^2}{2}} du
lim
N
→
∞
(
N
2
N
)
A
N
(
x
)
=
2
π
1
∫
−
∞
∞
e
−
2
u
2
d
u
.(N. 16)
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