π == 2t + 2Sum[(Cosh[n*t]*(Cos[t]*Sech[t])^n*Sin[n*t])/n, {n,∞}], (1≤t<2 ?) Is anybody playing with this? I find it incredible. Is it one of those Borwein or Zagier type high precision frauds? Bibasic telescopy? Am I missing something completely obvious? Inconclusive plotting suggests it holds in a region of the complex plane shaped like a football silhouette centered at t = π/2 + 0i of width > 1 and height < i. (DIY or stay tuned.) —rwg On Tue, Aug 28, 2018 at 7:29 PM Bill Gosper <billgosper@gmail.com> wrote:
On Tue, Aug 28, 2018 at 2:23 PM françois mendzina essomba2 < m_essob@yahoo.fr> wrote:
Hello,
Here is a identity that j found, with a ratio between cosine and hyperbolic cosine.
Pi=2*mu*x+2*sum((1/n)*(cos(mu*x)/cosh(alpha*x))^n*cosh(mu*n*x)*sin(alpha*n*x),n=1..infinity);
This identity seems true for the following conditions:
mu = alpha and x in [1/mu , 2/mu]
The value of x, for which this identity is the most convergent is close to :
x = (1.5)*(1/mu)
Umm, would you believe π/(2𝜇 ?-)
All smirking aside, for general t, e.g., Pi == 2*t + 2*Sum[(Cosh[n*t]*(Cos[t]*Sech[t])^n*Sin[n*t])/n, {n, Infinity}] (t≠π/2) I don't recall anything like it. OtOH, at my age, I don't recall a lot of things. --rwg
Note, as usual, the n=0 term would explain the prepended term if we took a limit for 0/0. Maybe we should redefine Sum to always take limits.
The values of x, verifying the identity seem to form a parable whose
vertex corresponds to (1/2) * (1 / mu), which I found curious0
Why ???