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It is well known that:

$\displaystyle \mathcal{H}_n = \log{n} + \gamma - \sum_{k=1}^\infty \frac{B_{k}}{k \, n^{k}}$

Where $\displaystyle B_k$ are the Bernoulli Numbers

A similar asymptotic expansion for the Harmonic Numbers begins with the logarithm shifted slightly:

$\displaystyle \mathcal{H}_n = \log{\left(n+\frac{1}{2}\right)} + \gamma + \sum_{k=2}^\infty \frac{A_k}{x^k}$

Is there a closed form for $\displaystyle A_k$ in terms of known number sequences?

The first few terms of $\displaystyle A_k$ are:

$\displaystyle \frac{1}{24}, -\frac{1}{24}, \frac{23}{960}, -\frac{1}{160}, -\frac{11}{8064}, -\frac{1}{896}, \frac{143}{30720}, -\frac{1}{4608}$

for $\displaystyle k \in \left[2,\cdots,9 \right]$

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Simply write

$$\begin{align} \log\left(n+\frac12\right)&=\log(n)+\log\left(1+\frac1{2n}\right)\\\\ &=\log(n)+\sum_{k=1}^\infty \frac{(-1)^{k-1}}{2^kk\,n^k} \end{align}$$

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  • $\begingroup$ Your answer is fine :) $\endgroup$ – Jack Tiger Lam Dec 9 '16 at 8:32

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