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Putnam
1969 Putnam
B5
B5
Part of
1969 Putnam
Problems
(1)
Putnam 1969 B5
Source: Putnam 1969
3/31/2022
Let
a
1
<
a
2
<
…
a_1 <a_2 < \ldots
a
1
<
a
2
<
…
be an increasing sequence of positive integers. Let the series
∑
i
=
1
∞
1
a
i
\sum_{i=1}^{\infty} \frac{1}{a_i }
i
=
1
∑
∞
a
i
1
be convergent. For any real number
x
x
x
, let
k
(
x
)
k(x)
k
(
x
)
be the number of the
a
i
a_i
a
i
which do not exceed
x
x
x
. Show that
lim
x
→
∞
k
(
x
)
x
=
0.
\lim_{x\to \infty} \frac{k(x)}{x}=0.
lim
x
→
∞
x
k
(
x
)
=
0.
Putnam
Sequences