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Putnam
1955 Putnam
A3
A3
Part of
1955 Putnam
Problems
(1)
Putnam 1955 A3
Source:
5/23/2022
Suppose that
∑
i
=
1
∞
x
i
\sum^\infty_{i=1} x_i
∑
i
=
1
∞
x
i
is a convergent series of positive terms which monotonically decrease (that is,
x
1
≥
x
2
≥
x
3
≥
⋯
x_1 \ge x_2 \ge x_3 \ge \cdots
x
1
≥
x
2
≥
x
3
≥
⋯
). Let
P
P
P
denote the set of all numbers which are sums of some (finite or infinite) subseries of
∑
i
=
1
∞
x
i
.
\sum^\infty_{i= 1} x_i.
∑
i
=
1
∞
x
i
.
Show that
P
P
P
is an interval if and only if
x
n
≤
∑
i
=
n
+
1
∞
x
i
x_n \le \sum^\infty_{i = n + 1} x_i
x
n
≤
i
=
n
+
1
∑
∞
x
i
for every integer
n
.
n.
n
.
Putnam