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t = lim sum_{j=1}^N q^{n_j}.

Source: Polish MO Second Round 1974 p5

September 8, 2024
limitalgebraSequence

Problem Statement

The given numbers are real numbers q,t∈⟨12;1) q,t \in \langle \frac{1}{2}; 1) , t∈(0;1⟩ t \in (0; 1 \rangle . Prove that there is an increasing sequence of natural numbers nk {n_k} (k=1,2,… k = 1,2, \ldots ) such that t=lim⁡N→∞∑j=1Nqnj. t = \lim_{N\to \infty} \sum_{j=1}^N q^{n_j}.
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