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Putnam 1955 A3

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May 23, 2022
Putnam

Problem Statement

Suppose that ∑i=1∞xi\sum^\infty_{i=1} x_i is a convergent series of positive terms which monotonically decrease (that is, x1≥x2≥x3≥⋯x_1 \ge x_2 \ge x_3 \ge \cdots). Let PP denote the set of all numbers which are sums of some (finite or infinite) subseries of ∑i=1∞xi.\sum^\infty_{i= 1} x_i. Show that PP is an interval if and only if xn≤∑i=n+1∞xi x_n \le \sum^\infty_{i = n + 1} x_i for every integer n.n.
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