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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 18
Prove the value of the limit is 0
Prove the value of the limit is 0
Source: 2019 Jozsef Wildt International Math Competition-W. 18
May 18, 2020
limit
Sequences
Problem Statement
Let
{
c
k
}
k
≥
1
\{c_k\}_{k\geq1}
{
c
k
}
k
≥
1
be a sequence with
0
≤
c
k
≤
1
0 \leq c_k \leq 1
0
≤
c
k
≤
1
,
c
1
≠
0
c_1 \neq 0
c
1
=
0
,
α
>
1
\alpha > 1
α
>
1
. Let
C
n
=
c
1
+
⋯
+
c
n
C_n = c_1 + \cdots + c_n
C
n
=
c
1
+
⋯
+
c
n
. Prove
lim
n
→
∞
C
1
α
+
⋯
+
C
n
α
(
C
1
+
⋯
+
C
n
)
α
=
0
\lim \limits_{n \to \infty}\frac{C_1^{\alpha}+\cdots+C_n^{\alpha}}{\left(C_1+\cdots +C_n\right)^{\alpha}}=0
n
→
∞
lim
(
C
1
+
⋯
+
C
n
)
α
C
1
α
+
⋯
+
C
n
α
=
0
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