Evaluate $\sum_{n=0}^\infty (-1)^n \ln\frac{2+2n}{1+2n}$

We have the series expansion of the digamma function:

$$\psi(z)=-\gamma+\sum_{n=0}^\infty\frac1{n+1}-\frac1{n+z}$$

Integrating both sides gives

$$\ln\Gamma(z)=(1-z)\gamma+\sum_{n=0}^\infty\frac{z-1}{n+1}+\ln\left[\frac{n+1}{n+z}\right]$$

Hence we have

$$\ln\Gamma(1)+\ln\Gamma(1/4)-\ln\Gamma(1/2)-\ln\Gamma(3/4)=\sum_{n=0}^\infty\ln\left[\frac{(n+1/2)(n+3/4)}{(n+1)(n+1/4)}\right]$$

$$\ln\left[\frac{\Gamma(1/4)}{\Gamma(3/4)}\right]-\frac12\ln(\pi)=\sum_{n=0}^\infty\ln\left[\frac{(4n+2)(4n+3)}{(4n+4)(4n+1)}\right]$$

Just want to point out that, a general version of this exist (Almodavar, Hall et.al https://cs.uwaterloo.ca/journals/JIS/VOL19/Moll/moll3.pdf).

\begin{eqnarray*} \prod_{n=0}^{\infty}{\left(\frac{2\alpha n+\beta}{2\alpha n+\gamma}\right)^{(-1)^{n}}} &=& 2^{\frac{\gamma-\beta}{2\alpha}} \frac{\Gamma^{2}\left(\frac{\gamma}{4\alpha}\right)}{\Gamma^{2}\left(\frac{\beta}{4\alpha}\right)} \frac{\Gamma\left(\frac{\beta}{2\alpha}\right)}{\Gamma\left(\frac{\gamma}{2\alpha}\right)} \end{eqnarray*}

Taking $\ln$ then will have,

\begin{eqnarray*} \sum_{n=0}^{\infty}(-1)^{n} {\ln \left(\frac{2\alpha n+\beta}{2\alpha n+\gamma}\right)} &=& \sum_{n=0}^{\infty}{\ln \left(\frac{2\alpha n+\beta}{2\alpha n+\gamma}\right)^{(-1)^{n}}} \\ &=& \ln \prod_{n=0}^{\infty}{\left(\frac{2\alpha n+\beta}{2\alpha n+\gamma}\right)^{(-1)^{n}}} \\ &=& \ln \left(2^{\frac{\gamma-\beta}{2\alpha}} \frac{\Gamma^{2}\left(\frac{\gamma}{4\alpha}\right)}{\Gamma^{2}\left(\frac{\beta}{4\alpha}\right)} \frac{\Gamma\left(\frac{\beta}{2\alpha}\right)}{\Gamma\left(\frac{\gamma}{2\alpha}\right)}\right) \end{eqnarray*}

The question of interest here is a case of $\alpha=1,\beta=2,$ and $\gamma=1$, which will then become,

\begin{eqnarray*} \sum_{n=0}^{\infty}(-1)^{n} {\ln \left(\frac{2 n+2}{2 n+1}\right)} &=& \ln \left(2^{-\frac{1}{2}} \frac{\Gamma^{2}\left(\frac{1}{4}\right)}{\Gamma^{2}\left(\frac{1}{2}\right)} \frac{\Gamma\left(1\right)}{\Gamma\left(\frac{1}{2}\right)}\right) \\ &=& \ln \left( \frac{1}{\sqrt{2 \pi}}\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{\pi} \right) \\ &\approx& 0.512377 \end{eqnarray*}

This is the same result as what is proved neatly by SimplyBeautifulArt. Recall that $\Gamma\left(\frac{3}{4}\right)=\frac{\sqrt{2}\pi }{\Gamma\left(\frac{1}{4}\right)}$.