olymmath

Putnam 1955 A3

  • Session A
  1. Suppose that ∑i=1∞xi\sum_{i=1}^{\infty} x_{i} is a convergent series of positive terms which monotonically decrease (that is, x1≥x2≥x3≥⋯x_{1} \geq x_{2} \geq x_{3} \geq \cdots ). Let PP denote the set of all numbers which are sums of some (finite or infinite) subseries of ∑i=1∞xi\sum_{i=1}^{\infty} x_{i}. Show that PP is an interval if and only if xn≤∑i=n+1∞xi for every integer n. (page 403) x_{n} \leq \sum_{i=n+1}^{\infty} x_{i} \text { for every integer } n . \text { (page 403) }

Dataset text PutnamGAP. Quoted with source.

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