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South Korea USCM
2019 Korea USCM
7
7
Part of
2019 Korea USCM
Problems
(1)
Condition for a series to converge
Source: 2019 South Korea USMC P7
8/13/2020
For a real number
a
a
a
and an integer
n
(
≥
2
)
n(\geq 2)
n
(
≥
2
)
, define
S
n
(
a
)
=
n
a
∑
k
=
1
n
−
1
1
k
2019
(
n
−
k
)
2019
S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}
S
n
(
a
)
=
n
a
k
=
1
∑
n
−
1
k
2019
(
n
−
k
)
2019
1
Find every value of
a
a
a
s.t. sequence
{
S
n
(
a
)
}
n
≥
2
\{S_n(a)\}_{n\geq 2}
{
S
n
(
a
)
}
n
≥
2
converges to a positive real.
analysis
series
college contests