What is the rate of growth of the partial sums of the reciprocals of the odd numbers?
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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 – Aryabhata Dec 13 '10 at 21:27
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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 $n$.
Barry Smith
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Thank you, Barry! $\gamma$ means Euler–Mascheroni constant? – Oleksandr Bondarenko Dec 13 '10 at 19:58
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