Evaluate the following sum
\(\displaystyle\sum_{m=1}^\infty \sum_{\substack{n\ge 1\\ (m,n)=1}} \frac{x^{m-1}y^{n-1}}{1-x^m y^n}\)
when |x|, |y|<1.
(We write (m,n) to denote the g.c.d of m and n.)
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Tag Archives: series
2009-15 Double sum
What is the value of the following infinite series?
\(\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}\)
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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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