Taylor's theorem

Series related to the power series of $\arcsin(x)$

We show the proof of the following result. The proof relies on the series expansion of arcsin. \[ \sum_{n=1}^{\infty} \frac{(2n-2)!}{2^{2n}(n!)^2 } = 1-\ln(2) \] Proof

Lets start with the expansion series of $\arcsin(x)$: \[ \sum_{n=0}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2 (2n+1)}x^{2n+1} = \arcsin(x) \quad \quad |x|\leq 1 \] Taking the derivative \[ \sum_{n=0}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2 }x^{2n} = \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}} \] Multiplying by $\frac{1}{x^2}$ and subtracting $-\frac{1}{x^2}$ \[ \sum_{n=1}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2 }x^{2n-2} = \frac{1}{x^{2}\sqrt{1-x^2}} -\frac{1}{x^2} \] Integrating \[ \int_{0}^{x} \sum_{n=1}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2 }x^{2n-2} dx = \sum_{n=1}^{\infty} \frac{(2n-2)!(2n)}{2^{2n}(n!)^2 }x^{2n-1}= \int_{0}^{x} \left(\frac{1}{x^{2}\sqrt{1-x^2}} -\frac{1}{x^2} \right)dx \] Hence \[ \int_{0}^{x} \left(\frac{1}{x^{2}\sqrt{1-x^2}} -\frac{1}{x^2} \right)dx = \left[ -\frac{\sqrt{1-x^2}}{x} +\frac{1}{x} \right]_{0}^{x} = -\frac{\sqrt{1-x^2}}{x} +\frac{1}{x}\] Therefore \[\sum_{n=1}^{\infty} \frac{(2n-2)!(2n)}{2^{2n}(n!)^2 }x^{2n-1}= -\frac{\sqrt{1-x^2}}{x} +\frac{1}{x} \] Integrating one more time from 0 to 1: \[\int_{0}^{1}\sum_{n=1}^{\infty} \frac{(2n-2)!(2n)}{2^{2n}(n!)^2 }x^{2n-1} dx=\sum_{n=1}^{\infty} \frac{(2n-2)!}{2^{2n}(n!)^2 }= \int_{0}^{1} \left(-\frac{\sqrt{1-x^2}}{x} +\frac{1}{x} \right)dx \] If $x=\sin(w)$ and $dx =\cos(w)dw$ \[ \int_{0}^{1} \left(-\frac{\sqrt{1-x^2}}{x} +\frac{1}{x}\right)dx = \int_{0}^{\frac{\pi}{2}} \left[ \cot(w) - \cot(w)\cos(w)\right]dw \] Using the following identities \[ \cos(w)\cot(w) = \csc(w)-\sin(w), \quad \cot(w)-\csc(x) = -\tan\left(\frac{w}{2}\right) \quad \textrm{and} \quad \sin(w)=2\sin\left(\frac{w}{2}\right)\cos\left(\frac{w}{2}\right)\] \begin{align*} \sum_{n=1}^{\infty} \frac{(2n-2)!}{2^{2n}(n!)^2 } =& \int_{0}^{\frac{\pi}{2}} \left[ \cot(w) - \cot(w)\cos(w)\right]dw \\ = & \int_{0}^{\frac{\pi}{2}} \left[ \cot(w) - \csc(w) +\sin(w) \right]dw \\ = & \int_{0}^{\frac{\pi}{2}} \left[ \cot(w) - \csc(w) +\sin(w) \right]dw \\ = & \int_{0}^{\frac{\pi}{2}} \left[ -\tan\left(\frac{w}{2}\right) +2\sin\left(\frac{w}{2}\right)\cos\left(\frac{w}{2}\right) \right]dw \\ = & \left[ 2\ln(\cos\left(\frac{x}{2}\right)) -2\cos^{2}\left(\frac{x}{2}\right) \right]_{0}^{\frac{\pi}{2}} = 1-\ln(2) \end{align*} Therefore \[ \boxed{\sum_{n=1}^{\infty} \frac{(2n-2)!}{2^{2n}(n!)^2 } = 1-\ln(2) }\]