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

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?

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

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

• Letting $x=1$ in $\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$ and $\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots$ the results are immediate. – L. F. Jul 26 '13 at 3:09
• Note that $\dfrac{\ln 4}{4}=\dfrac{\ln 2}{2}$, and $\frac12-\frac14+\frac16-\frac18+\cdots=\frac12\left(1-\frac12+\frac13-\frac14+ \cdots \right)$. – Jonas Meyer Jul 26 '13 at 3:16

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.