A series converging (or not) to $\mathrm{ln}2$

I have come across the following series, which I suspect converges to $\mathrm{ln}2$

$\sum _{k=1}^{\mathrm{\infty}}\frac{1}{{4}^{k}(2k)}{\textstyle (}\genfrac{}{}{0ex}{}{2k}{k}{\textstyle )}.$

I could not derive this series from some of the standard expressions for $\mathrm{ln}2$. The sum of the first

$100000$ terms agrees with $\mathrm{ln}2$ only up to two digits.

Does the series converge to $\mathrm{ln}2$?

I have come across the following series, which I suspect converges to $\mathrm{ln}2$

$\sum _{k=1}^{\mathrm{\infty}}\frac{1}{{4}^{k}(2k)}{\textstyle (}\genfrac{}{}{0ex}{}{2k}{k}{\textstyle )}.$

I could not derive this series from some of the standard expressions for $\mathrm{ln}2$. The sum of the first

$100000$ terms agrees with $\mathrm{ln}2$ only up to two digits.

Does the series converge to $\mathrm{ln}2$?