Power Series Expansion for Reciprocal of 1-z to the m+1 by Logarithm of Reciprocal of 1-z

Theorem

$\dfrac 1 {\paren {1 - z}^{m + 1} } \map \ln {\dfrac 1 {1 - z} } = \ds \sum_{k \mathop \ge 1} \paren {H_{m + k} - H_m} \dbinom {m + k} k z^k$

where:

$\dbinom {m + k} k$ denotes a binomial coefficient

$H_m$ denotes the $m$th harmonic number.

Proof

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Sources

• 1976: Derek A. Zave: A series expansion involving the harmonic numbers (Inf. Proc. Letters Vol. 5: pp. 75 – 77)
• 1990: Jürgen Spieß: Some Identities Involving Harmonic Numbers (Math. Comp. Vol. 55: pp. 839 – 863)  www.jstor.org/stable/2008451
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(25)$