Power Series Expansion for Logarithm of Tangent of x

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Theorem

\(\ds \ln \size {\tan x}\) \(=\) \(\ds \ln \size x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}\)
\(\ds \) \(=\) \(\ds \ln \size x + \frac {x^2} 3 + \frac {7 x^4} {90} + \frac {62 x^6} {2835} + \cdots\)


for all $x \in \R$ such that $0 < \size x < \dfrac \pi 2$.


Proof

From Power Series Expansion for Cosecant Function:

\(\ds 2 \csc 2 x\) \(=\) \(\ds 2 \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {2 x}^{2 n - 1} } {\paren {2 n}!}\)
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 2 \csc 2 x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2^{2 n + 1} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}\)

for $0 < \size {2 x} < \pi$.


From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:

\(\ds \int 2 \csc 2 x \rd x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \int \dfrac {\paren {-1}^{n - 1} 2^{2 n + 1} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!} \rd x\)
\(\ds \int 2 \csc 2 x \rd x\) \(=\) \(\ds \int \frac 1 x \rd x + \sum_{n \mathop = 1}^\infty \int \dfrac {\paren {-1}^{n - 1} 2^{2 n + 1} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!} \rd x\) extracting the zeroth term
\(\ds \leadsto \ \ \) \(\ds \ln \size {\tan x}\) \(=\) \(\ds \ln \size x + \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n - 1} 2^{2 n + 1} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n} } {\paren {2 n} \paren {2 n}!}\) Primitive of $\csc a x$, Integral of Power, Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \ln \size x + \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}\) simplification

for $0 < \size x < \dfrac \pi 2$.

$\blacksquare$


Also presented as

The Power Series Expansion for Logarithm of Tangent of x can also be presented in the form:

\(\ds \ln \size {\tan x}\) \(=\) \(\ds \ln \size x + \sum_{n \mathop = 1}^\infty \frac {2^{2 n} \paren {2^{2 n - 1} - 1} {B_n}^* x^{2 n} } {n \paren {2 n}!}\)

where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.


Sources

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