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Putnam
1965 Putnam
A3
Putnam 1965 A3
Putnam 1965 A3
Source:
September 27, 2020
Putnam
limit
Problem Statement
Show that, for any sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
of real numbers, the two conditions
lim
n
→
∞
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
)
n
=
α
\lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha
n
→
∞
lim
n
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
)
=
α
and
lim
n
→
∞
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
2
)
n
2
=
α
\lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha
n
→
∞
lim
n
2
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
2
)
=
α
are equivalent.
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