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South Korea USCM
2016 Korea USCM
2
2
Part of
2016 Korea USCM
Problems
(1)
decreasing seq, upper bound condition on partial sum, show series upper bound
Source: 2016 South Korea USCM P2
8/16/2020
Suppose
{
a
n
}
\{a_n\}
{
a
n
}
is a decreasing sequence of reals and
lim
n
→
∞
a
n
=
0
\lim\limits_{n\to\infty} a_n = 0
n
→
∞
lim
a
n
=
0
. If
S
2
k
−
2
k
a
2
k
≤
1
S_{2^k} - 2^k a_{2^k} \leq 1
S
2
k
−
2
k
a
2
k
≤
1
for any positive integer
k
k
k
, show that
∑
n
=
1
∞
a
n
≤
1
\sum_{n=1}^{\infty} a_n \leq 1
n
=
1
∑
∞
a
n
≤
1
(At here,
S
m
=
∑
n
=
1
m
a
n
S_m = \sum_{n=1}^m a_n
S
m
=
∑
n
=
1
m
a
n
is a partial sum of
{
a
n
}
\{a_n\}
{
a
n
}
.)
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