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Vojtěch Jarník IMC
2012 VJIMC
Problem 1
given limsup, integral converges
given limsup, integral converges
Source: VJIMC 2012 2.1
May 31, 2021
real analysis
calculus
integration
Problem Statement
Let
f
:
[
1
,
∞
)
→
(
0
,
∞
)
f:[1,\infty)\to(0,\infty)
f
:
[
1
,
∞
)
→
(
0
,
∞
)
be a non-increasing function such that
lim sup
n
→
∞
f
(
2
n
+
1
)
f
(
2
n
)
<
1
2
.
\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.
n
→
∞
lim
sup
f
(
2
n
)
f
(
2
n
+
1
)
<
2
1
.
Prove that
∫
1
∞
f
(
x
)
d
x
<
∞
.
\int^\infty_1f(x)\text dx<\infty.
∫
1
∞
f
(
x
)
d
x
<
∞.
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