How to evaluate the sum $\displaystyle\sum \limits_{n=1}^\infty \frac1{L_n}$? Where $L_n$ is the least common multiple of $1, 2, 3,\ldots, n$, e.g. least common multiple of $(6,3,4)$ is $12$.
Evaluate $\sum \limits_{n=1}^\infty \frac1{L_n}$ where $L_n$ is least common multiple of $1, 2, 3,\ldots,n$
11
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real-analysis
sequences-and-series
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0...neither is the reciprocal, $0.5593527977931594274755\dots$ – 2012-02-05
1 Answers
1
Convergence follows from $LCM(1,2,…,n)≥n(n−1)$ and $\sum_{n=2}^{\infty} \frac{1}{n(n-1)} < \infty$