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CyborgTosser (talk | contribs) sets |
CyborgTosser (talk | contribs) fxd formula, explain series expansion of ln(1+u) |
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Line 41:
& = & \exp \left ( \ln \prod_{n=1}^{\infty}(1 + z^{n})^{B_{n}} \right ) \\
& = & \exp \left ( \sum_{n = 1}^{\infty} B_{n} \ln(1 + z^{n}) \right ) \\
& = & \exp \left ( \sum_{n = 1}^{\infty} B_{n} \cdot \sum_{k = 1}^{\infty} \frac{(-1)^{k-1}z^{nk}}{k} \right ) \\
& = & \exp \left ( \sum_{k = 1}^{\infty} \frac{(-1)^{k-1}}{k} \cdot \sum_{n = 1}^{\infty}B_{n}z^{nk} \right ) \\
& = & \exp \left ( \sum_{k = 1}^{\infty} \frac{(-1)^{k-1} B(z^{k})}{k} \right )
\end{matrix}</math>
where the expansion
:<math>\ln(1 + u) = \sum_{k = 1}^{\infty} \frac{(-1)^{k-1}u^{k}}{k} </math> was used to go from line 4 to line 5.
[[Category:combinatorics]]
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