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Miklós Schweitzer
2020 Miklós Schweitzer
7
sequence bounded iff series converges
sequence bounded iff series converges
Source: Miklos Schweitzer 2020, Problem 7
December 1, 2020
analysis
real analysis
Problem Statement
Let
p
(
n
)
≥
0
p(n)\geq 0
p
(
n
)
≥
0
for all positive integers
n
n
n
. Furthermore,
x
(
0
)
=
0
,
v
(
0
)
=
1
x(0)=0, v(0)=1
x
(
0
)
=
0
,
v
(
0
)
=
1
, and
x
(
n
)
=
x
(
n
−
1
)
+
v
(
n
−
1
)
,
v
(
n
)
=
v
(
n
−
1
)
−
p
(
n
)
x
(
n
)
(
n
=
1
,
2
,
…
)
.
x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).
x
(
n
)
=
x
(
n
−
1
)
+
v
(
n
−
1
)
,
v
(
n
)
=
v
(
n
−
1
)
−
p
(
n
)
x
(
n
)
(
n
=
1
,
2
,
…
)
.
Assume that
v
(
n
)
→
0
v(n)\to 0
v
(
n
)
→
0
in a decreasing manner as
n
→
∞
n \to \infty
n
→
∞
. Prove that the sequence
x
(
n
)
x(n)
x
(
n
)
is bounded if and only if
∑
n
=
1
∞
n
⋅
p
(
n
)
<
∞
\sum_{n=1}^{\infty}n\cdot p(n)<\infty
∑
n
=
1
∞
n
⋅
p
(
n
)
<
∞
.
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