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1983 IMO Longlists
46
46
Part of
1983 IMO Longlists
Problems
(1)
Prove that u_n \to \infty [IMO Longlist 1983]
Source: IMO Longlist 1983
10/7/2010
Let
f
f
f
be a real-valued function defined on
I
=
(
0
,
+
∞
)
I = (0,+\infty)
I
=
(
0
,
+
∞
)
and having no zeros on
I
I
I
. Suppose that
lim
x
→
+
∞
f
′
(
x
)
f
(
x
)
=
+
∞
.
\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.
x
→
+
∞
lim
f
(
x
)
f
′
(
x
)
=
+
∞.
For the sequence
u
n
=
ln
∣
f
(
n
+
1
)
f
(
n
)
∣
u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|
u
n
=
ln
f
(
n
)
f
(
n
+
1
)
, prove that
u
n
→
+
∞
u_n \to +\infty
u
n
→
+
∞
as
n
→
+
∞
.
n \to +\infty.
n
→
+
∞.
function
limit
logarithms
algebra unsolved
algebra