2008-2 Strange representation (9/11)

Prove that if x is a real number such that \(0<x\le \frac12\), then x can be represented as an infinite sum

\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\),

where each \(n_k\) is an integer such that \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\).

x가 \(0<x\le \frac12\)을 만족하는 실수일때, x는 아래와 같은 무한급수로 표현할 수 있음을 보여라.

\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\).

여기서 각 \(n_k\)는 정수이며 \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\)을 만족한다.

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