$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Question

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$

Are there any known relations between ln(4) and pi, and also, have these alternating sums been discovered before?

$\frac42-\frac44+\frac46-\frac48+\frac4{10}-\frac4{12}+\frac4{14}\dots = \ln4$

$\frac41-\frac43+\frac45-\frac47+\frac49-\frac4{11}+\frac4{13}\dots = \pi$

Answer 1

Both were known to Leibniz in 1600's and are the integrals (from $0$ to $1$) of the geometric series for $\frac{1}{1+x}$ and $\frac{1}{1+x^2}$ respectively.