Timeline for Fast series for pi
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 24, 2017 at 22:16 | answer | added | Philip Thomas | timeline score: 0 | |
| Oct 29, 2016 at 17:16 | comment | added | Włodzimierz Holsztyński | I like the known iterating $\ x+\cos(x)\ $ till we get $\ 0=\cos(\pi).$ | |
| Apr 3, 2011 at 8:16 | history | edited | Seva |
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| Apr 2, 2011 at 15:04 | answer | added | Gert Almkvist | timeline score: 11 | |
| Feb 24, 2011 at 4:33 | vote | accept | Stanley Yao Xiao | ||
| Feb 23, 2011 at 1:04 | answer | added | Gerry Myerson | timeline score: 16 | |
| Feb 22, 2011 at 23:19 | comment | added | André Henriques | The question should be rephrased. Instead of "My first question is, what is the best known series for $\pi$? Namely, a series of the form above with the largest radius of convergence", I suggest: "Among all power series $f(x)=\sum_{n=0}^\infty a_n x^n$ with $f(1)=\pi$, and with the extra property that $a_{n+1}/a_n$ is a rational expression of $n$, which one has the largest radius of convergence?" | |
| Feb 22, 2011 at 22:30 | comment | added | Alex R. | Depends what you mean by best? If you want to find a specific digit, then something like the BBP formula should suit your needs. Otherwise the Chudnovsky modification of Ramanujan's formula is very fast. | |
| Feb 22, 2011 at 22:15 | comment | added | Gerry Myerson | But if $\sum a_nz^n$ has radius of convergence $r$, then $\sum b_nz^n$, where $b_n=a_n/2^n$, has radius of convergence $2r$, and doesn't do a better job of getting $\pi$. Are we assuming that $\pi$ is necessarily $f(1)$? | |
| Feb 22, 2011 at 22:11 | comment | added | Kevin O'Bryant | @Thierry: a large ROC means that $a_n\to0$ rapidly, so that the partial sums are good approximations. A more fundamental phrasing of the question would be to ask for the function $\alpha(n)$ that can be computed with the fewest flops (asymptotically) and which satisfies $|\pi-\alpha(n)|<1/n$. | |
| Feb 22, 2011 at 22:00 | comment | added | Thierry Zell | I'm not sure I get your question. Why look at the radius of convergence? | |
| Feb 22, 2011 at 21:46 | history | asked | Stanley Yao Xiao | CC BY-SA 2.5 |