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More analysis... Find a bound for sum of sqrt[k]{x}
Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 3
10/21/2023
Prove that there exists a constant
C
>
0
C > 0
C
>
0
such that, for any integers
m
,
n
m, n
m
,
n
with
n
≥
m
>
1
n \geq m > 1
n
≥
m
>
1
and any real number
x
>
1
x > 1
x
>
1
,
∑
k
=
m
n
x
k
≤
C
(
m
2
⋅
x
m
−
1
log
x
+
n
)
\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)
k
=
m
∑
n
k
x
≤
C
(
lo
g
x
m
2
⋅
m
−
1
x
+
n
)
real analysis
inequalities
algebra