Power Series Expansion for Real Area Hyperbolic Tangent

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Theorem

The (real) area hyperbolic tangent function has a Taylor series expansion:

\(\ds \artanh x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}\)
\(\ds \) \(=\) \(\ds x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots\)

for $\size x < 1$.


Proof

From Sum of Infinite Geometric Sequence:

$(1): \quad \ds \frac 1 {1 - x^2} = \sum_{n \mathop = 0}^\infty \paren {x^2}^n$

for $-1 < x < 1$.

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

\(\ds \int_0^x \frac 1 {1 - t^2} \rd t\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \int_0^x \paren {t^2}^n \rd t\)
\(\ds \leadsto \ \ \) \(\ds \artanh x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}\) Primitive of Reciprocal of $\dfrac 1 {1 - t^2}$, Integral of Power

For $-1 < x < 1$, the sequence $\sequence {\dfrac {x^{2 n + 1} } {2 n + 1} }$ is decreasing and converges to zero.

Hence the result.

$\blacksquare$


Also see


Sources

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