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KoMaL A Problems 2021/2022
A. 825
Functional equation with lim f(n+1)/f(n)=1
Functional equation with lim f(n+1)/f(n)=1
Source: KoMaL A. 825
May 11, 2022
algebra
functional equation
komal
Problem Statement
Find all functions
f
:
Z
+
→
R
+
f:\mathbb Z^+\to\mathbb R^+
f
:
Z
+
→
R
+
that satisfy
f
(
n
k
2
)
=
f
(
n
)
f
2
(
k
)
f(nk^2)=f(n)f^2(k)
f
(
n
k
2
)
=
f
(
n
)
f
2
(
k
)
for all positive integers
n
n
n
and
k
k
k
, furthermore
lim
n
→
∞
f
(
n
+
1
)
f
(
n
)
=
1
\lim\limits_{n\to\infty}\dfrac{f(n+1)}{f(n)}=1
n
→
∞
lim
f
(
n
)
f
(
n
+
1
)
=
1
.
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