3
$\begingroup$

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?

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

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

$\endgroup$
  • 5
    $\begingroup$ 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. $\endgroup$ – L. F. Jul 26 '13 at 3:09
  • 1
    $\begingroup$ Note that $\dfrac{\ln 4}{4}=\dfrac{\ln 2}{2}$, and $\frac12-\frac14+\frac16-\frac18+\cdots=\frac12\left(1-\frac12+\frac13-\frac14+ \cdots \right)$. $\endgroup$ – Jonas Meyer Jul 26 '13 at 3:16
4
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ @AlbertRenshaw: There is a connection. However, the connection that I know travels through the complex numbers, and is difficult to describe briefly. $\endgroup$ – André Nicolas Jul 26 '13 at 3:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.