One of three of a kind - Part (2)

Calculus Level 4

\[ \large \sum_{k=1}^\infty \frac{\mu(k)}k \ln(1 - x^k) \]

For \(0<x<1 \), find the closed form for the series above.

Note

\( \mu(n) \) denote the Möbius Function:

\[ \mu(n) = \begin{cases} 0 & \quad & \text{ if } n \text{ has more repeated prime factors} \\ 1 & \quad & \text{ if } n =1 \\ (-1)^n & \quad & \text{ if } n \text{ is a product of } k \text{ distinct primes} \end{cases}\]

Try part 1 and part 3

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