Variance of Logistic Distribution/Lemma 3

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Lemma for Variance of Logistic Distribution

$\ds \int_{\to 0}^{\to 1} \map \ln u \map \ln {1 - u} \rd u = 2 - \dfrac {\pi^2} 6$


Proof

From Corollary to Power Series Expansion for Logarithm of 1 + x we have:

$\ds \ln \paren {1 - x} = -\sum_{n \mathop = 1}^\infty \dfrac {x^n} n$

Therefore:

\(\ds \int_{\to 0}^{\to 1} \map \ln u \map \ln {1 - u} \rd u\) \(=\) \(\ds \int_{\to 0}^{\to 1} \map \ln u \paren {-\sum_{n \mathop = 1}^\infty \dfrac {u^n} n} \rd u\)
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \dfrac 1 n \int_{\to 0}^{\to 1} u^n \map \ln u \rd u\) Fubini's Theorem
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \dfrac 1 n \bigintlimits {\dfrac {u^{n + 1} } {n + 1} \paren {\ln u - \dfrac 1 {n + 1} } } 0 1\) Primitive of $u^n \ln u$
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \dfrac 1 n \frac 1 {n + 1} \paren {0 - \frac 1 {n + 1} }\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n \dfrac 1 {\paren {n + 1}^2}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n - \sum_{n \mathop = 1}^\infty \dfrac 1 {\paren {n + 1} } - \sum_{n \mathop = 1}^\infty \dfrac 1 {\paren {n + 1}^2}\) Partial Fractions Expansion of $\dfrac 1 {x \paren {x + 1}^2}$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n \paren {n + 1} } - \paren {\sum_{n \mathop = 1}^\infty \dfrac 1 {n^2} - 1}\)
\(\ds \) \(=\) \(\ds 1 - \paren {\dfrac {\pi^2} 6 - 1}\) Sum of Sequence of Reciprocals of Triangular Numbers and Basel Problem
\(\ds \) \(=\) \(\ds 2 - \dfrac {\pi^2} 6\)

$\blacksquare$

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