Sum over k from 2 to Infinity of Zeta of k Over k Alternating in Sign
Jump to navigation
Jump to search
Theorem
| \(\ds \sum_{k \mathop = 2}^\infty \dfrac {\paren {-1}^k} k \map \zeta k\) | \(=\) | \(\ds \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \gamma\) |
Proof 1
| \(\ds \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \map \zeta k x^{k - 1}\) | \(=\) | \(\ds \map H x\) | Power Series Expansion for Harmonic Numbers | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int_{\to 0}^{\to 1} \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \map \zeta k x^{k - 1} \rd x\) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \map H x \rd x\) | integrating with respect to $x$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 2}^\infty \dfrac {\paren {-1}^k} k \map \zeta k\) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \map H x \rd x\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - t^x} {1 - t} } \rd t \rd x\) | Reciprocal times Derivative of Gamma Function: Corollary and Extension of Harmonic Number to Non-Integer Argument | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - e^{x \ln t} } {1 - t} } \rd t \rd x\) | Logarithm of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} } \rd t \rd x - \int_{\to 0}^{\to 1} \int_{\to 0}^{\to 1} \paren {\dfrac {e^{x \ln t} } {1 - t} } \rd t \rd x\) | Linear Combination of Integrals | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} } \rd t \bigintlimits x 0 1 - \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} } \rd t \intlimits {\dfrac {e^{x \ln t} } {\ln t} } 0 1\) | Primitive of $e^{a x}$ and Primitive of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} } \rd t - \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} } \rd t \paren {\dfrac {t - 1} {\ln t} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} } \rd t + \int_{\to 0}^{\to 1} \paren {\dfrac 1 {\ln t} } \rd t\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \paren {\dfrac 1 {1 - t} + \dfrac 1 {\ln t} } \rd t\) | Linear Combination of Integrals | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_\infty^0 \paren {\dfrac 1 {1 - e^{-u} } + \dfrac 1 {\map \ln {e^{-u} } } } \paren {-e^{-u} \rd u}\) | $t \to e^{-u}$ and $\rd t \to -e^{-u} \rd u$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \paren {\dfrac {e^{-u} } {1 - e^{-u} } - \dfrac {e^{-u} } u} \rd u\) | Reversal of Limits of Definite Integral | |||||||||||
| \(\ds \) | \(=\) | \(\ds - \int_0^\infty \paren {\frac {e^{-u} } u - \frac {e^{- u} } {1 - e^{-u} } } \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\map \psi 1\) | Gauss's Integral Form of Digamma Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \gamma\) | Digamma Function of One |
$\blacksquare$
Proof 2
| \(\ds \map \ln {\map \Gamma {x + 1} }\) | \(=\) | \(\ds -\gamma x + \sum_{k \mathop = 2}^{\infty} \dfrac {\map \zeta k \paren {-x}^k} k\) | Power Series Expansion for Logarithm of Gamma Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\gamma x + \dfrac {\map \zeta 2 x^2} 2 - \dfrac {\map \zeta 3 x^3} 3 + \dfrac {\map \zeta 4 x^4} 4 - \dfrac {\map \zeta 5 x^5} 5 + \cdots\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \ln {\map \Gamma {1 + 1} }\) | \(=\) | \(\ds -\gamma + \sum_{k \mathop = 2}^{\infty} \dfrac {\map \zeta k \paren {-1}^k} k\) | setting $x = 1$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds -\gamma + \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots\) | $\map \Gamma 2 = 1$ and Natural Logarithm of 1 is 0 | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \gamma\) | \(=\) | \(\ds \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots\) | rearranging |
$\blacksquare$