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Lebl-Basic-Analysis-I
· Problem 2.6.14
Sequences and Series
Mathematics · power-series
Problem
Suppose
{
c
n
}
n
=
1
∞
\{c_n\}_{n=1}^{\infty}
{
c
n
}
n
=
1
∞
is a sequence. Prove that for every
r
∈
(
0
,
1
)
r \in (0,1)
r
∈
(
0
,
1
)
, there exists a strictly increasing sequence
{
n
k
}
k
=
1
∞
\{n_k\}_{k=1}^{\infty}
{
n
k
}
k
=
1
∞
of natural numbers (
n
k
+
1
>
n
k
n_{k+1} > n_k
n
k
+
1
>
n
k
) such that
∑
k
=
1
∞
c
k
x
n
k
\sum_{k=1}^{\infty} c_k x^{n_k}
k
=
1
∑
∞
c
k
x
n
k
converges absolutely for all
x
∈
[
−
r
,
r
]
x \in [-r, r]
x
∈
[
−
r
,
r
]
.
Related theory
power-series
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