Power Series Expansion for Real Area Hyperbolic Sine

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

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

We will now prove that the series converges for $-1 < x < 1$.

By Stirling's Formula:

Then:

Hence by Convergence of P-Series:


 * $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^{3/2} }$

is convergent.

So by the Comparison Test, the Taylor series is convergent for $-1 \le x \le 1$.

Another lemma:

Lemma $2$
Let $x \ge 1$.

Let $z = \dfrac 1 x$.

Then we have:
 * $0 < \dfrac 1 z \le 1$

Now we consider:

Now let $x \le -1$.

We have that Inverse Hyperbolic Sine is Odd Function.

That is:
 * $\arsinh x = -\map \arsinh {-x}$

Thus for $x \le -1$:

Hence the result.

Also presented as
This can also be presented as:

where $\pm$ is $+$ for $x \ge 1$ and $-$ for $x \le -1$.

Also see

 * Power Series Expansion for Real Area Hyperbolic Cosine
 * Power Series Expansion for Real Area Hyperbolic Tangent
 * Power Series Expansion for Real Area Hyperbolic Cotangent
 * Power Series Expansion for Real Area Hyperbolic Secant
 * Power Series Expansion for Real Area Hyperbolic Cosecant