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2000 Moldova National Olympiad
Problem 2
limit of binomial sequence
limit of binomial sequence
Source: Moldova 2000 Grade 12 P2
April 27, 2021
limit
real analysis
Problem Statement
For
n
∈
N
n\in\mathbb N
n
∈
N
, define
a
n
=
1
(
n
1
)
+
1
(
n
2
)
+
…
+
1
(
n
n
)
.
a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.
a
n
=
(
1
n
)
1
+
(
2
n
)
1
+
…
+
(
n
n
)
1
.
(a) Prove that the sequence
b
n
=
a
n
n
b_n=a_n^n
b
n
=
a
n
n
is convergent and determine the limit. (b) Show that
lim
n
→
∞
b
n
>
(
3
2
)
3
+
2
\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}
lim
n
→
∞
b
n
>
(
2
3
)
3
+
2
.
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