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National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1987 Flanders Math Olympiad
4
A nice limit
A nice limit
Source: flanders '87
September 24, 2005
limit
inequalities
Problem Statement
Show that for
p
>
1
p>1
p
>
1
we have
lim
n
→
+
∞
1
p
+
2
p
+
.
.
.
+
(
n
−
1
)
p
+
n
p
+
(
n
−
1
)
p
+
.
.
.
+
2
p
+
1
p
n
2
=
+
∞
\lim_{n\rightarrow+\infty}\frac{1^p+2^p+...+(n-1)^p+n^p+(n-1)^p+...+2^p+1^p}{n^2} = +\infty
n
→
+
∞
lim
n
2
1
p
+
2
p
+
...
+
(
n
−
1
)
p
+
n
p
+
(
n
−
1
)
p
+
...
+
2
p
+
1
p
=
+
∞
Find the limit if
p
=
1
p=1
p
=
1
.
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