Power Series Expansion for Harmonic Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

The harmonic numbers have the power series expansion:

\(\ds \map H x\) \(=\) \(\ds \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \map \zeta k x^{k - 1}\)
\(\ds \) \(=\) \(\ds \map \zeta 2 x - \map \zeta 3 x^2 + \map \zeta 4 x^3 - \map \zeta 5 x^4 + \cdots\)

where $\zeta$ denotes the Riemann zeta function.


This is valid for all $x \in \C$ such that $\size x < 1$.


Proof

\(\ds \map H x\) \(=\) \(\ds \gamma + \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }\) Extension of Harmonic Number to Non-Integer Argument
\(\ds \) \(=\) \(\ds \gamma - \gamma + \sum_{m \mathop = 1}^\infty \paren {\frac 1 m - \frac 1 {x + m} }\) Reciprocal times Derivative of Gamma Function
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\frac 1 m - \frac 1 {x + m} }\)
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\frac x m - \frac x {x + m} } \dfrac 1 x\) multiplying by $1$
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\frac x m + \paren {\frac m {x + m} - \frac {x + m} {x + m} } } \dfrac 1 x\)
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\frac x m + \paren {\frac m {x + m} - 1} } \dfrac 1 x\)
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\frac x m + \paren {\frac 1 {1 - \paren {-\dfrac x m } } } - 1} \dfrac 1 x\)
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\frac x m + \paren {\sum_{k \mathop = 0}^\infty \paren {-\dfrac x m}^k } - 1} \dfrac 1 x\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^\infty \paren {\sum_{k \mathop = 2}^\infty \paren {-\dfrac x m}^k } \dfrac 1 x\) reindexing the sum
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \paren {\sum_{m \mathop = 1}^\infty \dfrac 1 {m^k} } x^{k - 1}\) Fubini's Theorem
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \map \zeta k x^{k - 1}\) Definition of Riemann Zeta Function


Our reliance upon the Sum of Infinite Geometric Sequence above where $m = 1$ implies $\size x < 1$, where $\size x$ denotes the complex modulus of $x$.

When $m = 1$ and $x = 1$, we have Grandi's Series, a well known divergent series.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Power_Series_Expansion_for_Harmonic_Numbers&oldid=671124"