A familiar constant.......

This identity, when calculated correctly, is a very familiar constant. The question is this: how many accurate digits are there in its decimal expansion? {nl} Ln{[640,320^3 + 744]^2 - 393,768} / sqrt(652)=?

"\(\dots\) one example on Mathworld's Pi Approximations page.

The same formula from Ramanujan that I have used was extended in a different way by Warda (2004) with the result:

π   ≈  ln [ ( 640320³ + 744 )² - 2 · 196884 ] / ( 2 · \(\sqrt{163}\) )    [ Heureka: \(\pi \approx \frac{ \ln{ [~ ( 640320^3 + 744 )^2 - 2 \cdot 196884 ~]} } { 2 \cdot \sqrt{163}} \) ]

giving the resulting approximation to π :

π   ≈ 3.1415926535897932384626433832795028841971693992820...

that is accurate to 46 digits past the decimal point \(\dots \)".

( From http://members.bex.net/jtcullen515/math7.htm )

see: http://members.bex.net/jtcullen515/math7.htm

and see: http://mathworld.wolfram.com/PiApproximations.html

This evaluates to "pi"

Since this an irrational number, it is never precisely "accurate".....however....we can make it as "accurate" as we want to a specified number of digits......

Thank you CPhill: How many accurate decimal digits can you get? What I mean is this: 355/113=3.141592920.......gives you 6 accurate decimal digits, so with that expression, how many accurate decimal digits can you get out of it?

Oh....I see what you mean.....!!!

Here are the 1st 100 digits of pi :

3.[141592653589793238462643383279502884197169399]37510582097494459

23078164062862089986280348253421170679

Here is the result of your expression as given by WolframAlpha :

3.[141592653589793238462643383279502884197169399]2820711147894514138

916383017070856233260338755405432911712444501586...

So...it looks as though your expression is accurate  up to 45 decimal digits  [if i counted correctly.....!!!]

This expression gives:

3.1415926535 8979323846 2643383279 5028841971 69399 28207 1114789451 4138916383 0

Pi is:

3.1415926535 8979323846 2643383279 5028841971 69399 37510 5820974944 5923078164 

So, by simple counting, the expression in question gives 45 accurate decimal digits of Pi.