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Putnam
2010 Putnam
A6
Putnam 2010 A6
Putnam 2010 A6
Source:
December 6, 2010
Putnam
function
limit
integration
logarithms
calculus
inequalities
Problem Statement
Let
f
:
[
0
,
∞
)
→
R
f:[0,\infty)\to\mathbb{R}
f
:
[
0
,
∞
)
→
R
be a strictly decreasing continuous function such that
lim
x
→
∞
f
(
x
)
=
0.
\lim_{x\to\infty}f(x)=0.
lim
x
→
∞
f
(
x
)
=
0.
Prove that
∫
0
∞
f
(
x
)
−
f
(
x
+
1
)
f
(
x
)
d
x
\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx
∫
0
∞
f
(
x
)
f
(
x
)
−
f
(
x
+
1
)
d
x
diverges.
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