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Putnam
1960 Putnam
B6
Putnam 1960 B6
Putnam 1960 B6
Source: Putnam 1960
June 11, 2022
Putnam
p-adics
series
Problem Statement
Any positive integer
n
n
n
can be written in the form
n
=
2
k
(
2
l
+
1
)
n=2^{k}(2l+1)
n
=
2
k
(
2
l
+
1
)
with
k
,
l
k,l
k
,
l
positive integers. Let
a
n
=
e
−
k
a_n =e^{-k}
a
n
=
e
−
k
and
b
n
=
a
1
a
2
a
3
⋯
a
n
.
b_n = a_1 a_2 a_3 \cdots a_n.
b
n
=
a
1
a
2
a
3
⋯
a
n
.
Prove that
∑
n
=
1
∞
b
n
\sum_{n=1}^{\infty} b_n
n
=
1
∑
∞
b
n
converges.
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