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Putnam
2012 Putnam
4
Putnam 2012 B4
Putnam 2012 B4
Source:
December 3, 2012
Putnam
logarithms
limit
inequalities
induction
calculus
integration
Problem Statement
Suppose that
a
0
=
1
a_0=1
a
0
=
1
and that
a
n
+
1
=
a
n
+
e
−
a
n
a_{n+1}=a_n+e^{-a_n}
a
n
+
1
=
a
n
+
e
−
a
n
for
n
=
0
,
1
,
2
,
…
.
n=0,1,2,\dots.
n
=
0
,
1
,
2
,
…
.
Does
a
n
−
log
n
a_n-\log n
a
n
−
lo
g
n
have a finite limit as
n
→
∞
?
n\to\infty?
n
→
∞
?
(Here
log
n
=
log
e
n
=
ln
n
.
\log n=\log_en=\ln n.
lo
g
n
=
lo
g
e
n
=
ln
n
.
)
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