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2003 IMC
6
IMC 2003 Problem 12
IMC 2003 Problem 12
Source: IMC 2003 Day 2 Problem 6
November 2, 2020
limit
Convergence
real analysis
Problem Statement
Let
(
a
n
)
(a_{n})
(
a
n
)
be the sequence defined by
a
0
=
1
,
a
n
+
1
=
∑
k
=
0
n
a
k
n
−
k
+
2
a_{0}=1,a_{n+1}=\sum_{k=0}^{n}\dfrac{a_k}{n-k+2}
a
0
=
1
,
a
n
+
1
=
∑
k
=
0
n
n
−
k
+
2
a
k
. Find the limit
lim
n
→
∞
∑
k
=
0
n
a
k
2
k
,
\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\dfrac{a_{k}}{2^{k}},
n
→
∞
lim
k
=
0
∑
n
2
k
a
k
,
if it exists.
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