# The rate of growth of the partial sums of the reciprocals of the odd numbers

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

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