What is the rate of growth of the partial sums of the reciprocals of the odd numbers?
The rate of growth of the partial sums of the reciprocals of the odd numbers
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sequences-and-series
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1This was in fact shown and used recently here: http://math.stackexchange.com/questions/13888/how-to-sum-frac11-cdot-2-cdot-3-cdot-4-frac43-cdot-4-cdot-5-cdot-6-fr/13894#13894 – 2010-12-13
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0Thank you for helpful comment. – 2010-12-13
1 Answers
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$\sum_{1}^{n} \frac{1}{2i-1} = \sum_{1}^{2n} \frac{1}{i} - \frac{1}{2}\sum_{1}^{n} \frac{1}{i}$, and this is approximately $\ln(2n) - \frac{1}{2}\ln(n)+\frac{1}{2} \gamma = \frac{1}{2} \ln(n) + \ln(2) + \frac{1}{2} \gamma$ for large $n$.
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0Thank you, Barry! $\gamma$ means Euler–Mascheroni constant? – 2010-12-13
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0Yes, it does indeed. – 2010-12-13