Power Series Expansion for Logarithm of Cosine of x

Theorem

$\ds \ln \size {\cos x}$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}$

$\ds $

$=$

$\ds -\frac {x^2} 2 - \frac {x^4} {12} - \frac {x^6} {45} - \frac {17 x^8} {2520} - \cdots$

for all $x \in \R$ such that $\size x < \dfrac \pi 2$.

$\text {(1)}: \quad$

$\ds \tan x$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$

$\ds $

$=$

$\ds x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {315} + \cdots$

for $\size x < \dfrac \pi 2$.

$\ds \int_0^x \tan x \rd x$

$=$

$\ds \sum_{n \mathop = 0}^\infty \int_0^x \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!} \rd x$

$\ds \leadsto \ \ $

$\ds -\ln \size {\cos x}$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n} } {\paren {2 n} \paren {2 n}!}$

$\ds \leadsto \ \ $

$\ds \ln \size {\cos x}$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}$

simplification

$\blacksquare$