Summation of Power Series by Harmonic Sequence
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Theorem
Consider the power series:
- $\map f x = \ds \sum_{k \mathop \ge 0} a_k x^k$
Let $\map f x$ converge for $x = x_0$.
Then:
- $\ds \sum_{k \mathop \ge 0} a_k {x_0}^k H_k = \int_0^1 \dfrac {\map f {x_0} - \map f {x_0 y} } {1 - y} \rd y$
where $H_n$ denotes the $n$th harmonic number.
Proof
\(\ds \int_0^1 \dfrac {\map f {x_0} - \map f {x_0 y} } {1 - y} \rd y\) | \(=\) | \(\ds \int_0^1 \dfrac {\sum_{k \mathop \ge 0} a_k {x_0}^k - \sum_{k \mathop \ge 0} a_k {x_0}^k y^k} {1 - y} \rd y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} a_k {x_0}^k \int_0^1 \dfrac {1 - y^k} {1 - y} \rd y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} a_k {x_0}^k \int_0^1 \paren {1 + y + y^2 + \cdots + y^{k - 1} } \rd y\) | Sum of Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} a_k {x_0}^k \intlimits {y + \dfrac {y^2} 2 + \dfrac {y^3} 3 + \cdots + \dfrac {y^k} k} 0 1\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} a_k {x_0}^k \paren {\paren {1 + \dfrac {1^2} 2 + \dfrac {1^3} 3 + \cdots + \dfrac {1^k} k} - \paren {0 + \dfrac {0^2} 2 + \dfrac {0^3} 3 + \cdots + \dfrac {0^k} k} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} a_k {x_0}^k H_k\) | Definition of Harmonic Numbers |
$\blacksquare$
Sources
- 1961: H.W. Gould: Some Relations Involving the Finite Harmonic Series (Math. Mag. Vol. 34: pp. 317 – 321) www.jstor.org/stable/2688350
- 1962: N.R. Riesenberg and David Zeitlin: 4945 (Amer. Math. Monthly Vol. 69: p. 239) www.jstor.org/stable/2311070
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $20$