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Moldova Contests
Moldova National Olympiad
2003 Moldova National Olympiad
12.1
Intricate sequence
Intricate sequence
Source: Moldovan MO 2003
February 20, 2006
limit
logarithms
calculus
calculus computations
Problem Statement
For every natural number
n
n
n
let:
a
n
=
l
n
(
1
+
2
e
+
4
e
4
+
⋯
+
2
n
e
n
2
)
a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})
a
n
=
l
n
(
1
+
2
e
+
4
e
4
+
⋯
+
2
n
e
n
2
)
. Find:
lim
n
→
∞
a
n
n
2
\displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}}
n
→
∞
lim
n
2
a
n
.
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