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What is the rate of growth of the partial sums of the reciprocals of the odd numbers?

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This was in fact shown and used recently here: math.stackexchange.com/questions/13888/… –  Aryabhata Dec 13 '10 at 21:27
    
Thank you for helpful comment. –  Oleksandr Bondarenko Dec 13 '10 at 21:58

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up vote 6 down vote accepted

$\sum_{1}^{n} \frac{1}{2i-1} = \sum_{1}^{2n} \frac{1}{i} - \frac{1}{2}\sum_{1}^{n} \frac{1}{i}$, 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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Thank you, Barry! $\gamma$ means Euler–Mascheroni constant? –  Oleksandr Bondarenko Dec 13 '10 at 19:58
    
Yes, it does indeed. –  Barry Smith Dec 13 '10 at 20:01

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