Interesting Pi summation

[galactus]

Has anyone ever seen this one?.

\(\displaystyle \L\\\pi=\sum_{n=0}^{\infty}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)\cdot\left(\frac{1}{16}\right)^{n}\)

This is probably one of the most accurate and fastest converging.

I just thought it was interesting.

 

[Deleted member 4993]

That is fast!!

at n=2, S = 3.14158739 - all the accuracy I need.

 

[Denis]

In my high school days (late 50's), we used 22/7 ; 3.142

Only "calculators" then were noisy machines with cranks;
to multiply, you cranked a whole lotta additions

 

[stapel]

In my high school days (late 50's)... [the o]nly "calculators" then were noisy machines with cranks; to multiply, you cranked a whole lotta additions

And this made one feel crank-y...? :wink:

Eliz.

 

[galactus]

Speaking of old calculators, they have become quite the collectors item.
The early electronics, I mean. An HP35 with the 'bug' can bring $1000.
I seen an old Sanyo ICC-0081(circa 1971) go for $300 on eBay recently. One of the rarest is a TI-150. Very, very rare. Would bring a hefty price if you could find one. I have a Sanyo ICC 811 mini. Mini indeed. It's over 8 inches long. It's comical how primitive(Nixie tubes and only basic arithmetic) they were compared to what we have now. It all had to start somewhere.

If anyone is interested, see here:

http://www.vintagecalculators.com

 

[morson]

What do you mean by "most accurate"? Is the sum to infinity of that series actually equal to pi? If so, cool. After evaluating n = 0 I'm already at about 3.14 in my head. The series calculates pi to 2 decimal places after the sum of one term!