MathDB
Problems
Contests
Undergraduate contests
Putnam
1964 Putnam
A5
A5
Part of
1964 Putnam
Problems
(1)
Putnam 1964 A5
Source: Putnam 1964
3/5/2022
Prove that there exists a constant
K
K
K
such that the following inequality holds for any sequence of positive numbers
a
1
,
a
2
,
a
3
,
…
:
a_1 , a_2 , a_3 , \ldots:
a
1
,
a
2
,
a
3
,
…
:
∑
n
=
1
∞
n
a
1
+
a
2
+
…
+
a
n
≤
K
∑
n
=
1
∞
1
a
n
.
\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 +\ldots + a_n } \leq K \sum_{n=1}^{\infty} \frac{1}{a_{n}}.
n
=
1
∑
∞
a
1
+
a
2
+
…
+
a
n
n
≤
K
n
=
1
∑
∞
a
n
1
.
Putnam
inequalities