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Miklós Schweitzer
1958 Miklós Schweitzer
7
Miklós Schweitzer 1958- Problem 7
Miklós Schweitzer 1958- Problem 7
Source:
October 23, 2015
college contests
Problem Statement
7. Let
a
0
a_0
a
0
and
a
1
a_1
a
1
be arbitrary real numbers, and let
a
n
+
1
=
a
n
+
2
n
+
1
a
n
−
1
a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}
a
n
+
1
=
a
n
+
n
+
1
2
a
n
−
1
(
n
=
1
,
2
,
…
)
(n= 1, 2, \dots)
(
n
=
1
,
2
,
…
)
Show that the sequence
(
a
n
n
2
)
n
=
1
∞
\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}
(
n
2
a
n
)
n
=
1
∞
is convergent and find its limit. (S. 10)
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