The Value of Tau

To find the exact value of (at least to a certain number of decimal places) we can take the easy way and write a short C program, TAU.C. The output of the program is:

The number eponymous with the Greek letter tauis defined as

Thus even though is defined by an expression which involves the unlimited addition of increasingly large integers it is nevertheless a finite quantity, namely, 1.7579327566180045 (to 16 decimal places) — another mathematical oddity.This program calculates tau to 16 decimal places thus: tau = sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... + sqrt(22)))...))) 1.5537739740300374 = sqrt(1+sqrt(2)) 1.7122650649295326 = sqrt(1+sqrt(2+sqrt(3))) 1.7487627132551438 = sqrt(1+sqrt(2+sqrt(3+sqrt(4)))) 1.7562384875823431 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5))))) 1.7576412350415822 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+sqrt(6)))))) 1.7578856460964372 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+sqrt(6+sqrt(7))))))) 1.7579255575682606 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+...+sqrt(8)))))))) 1.7579317105145658 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+...+sqrt(9))))))))) 1.7579326113938309 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+...+sqrt(10)))))))))) 1.7579327372820677 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+...+sqrt(11))))))))))) 1.7579327541406276 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+...+sqrt(12)))))))))))) 1.7579327563117013 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(13))))))))))))) 1.7579327565813685 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(14)))))))))))))) 1.7579327566137564 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(15))))))))))))))) 1.7579327566175260 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(16)))))))))))))))) 1.7579327566179521 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(17))))))))))))))))) 1.7579327566179990 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(18)))))))))))))))))) 1.7579327566180039 = sqrt(1+sqrt(2+sqrt(3+sqrt(4+...+sqrt(19))))))))))))))))))) 1.7579327566180045 = sqrt(1+sqrt(2+sqrt(3+...+sqrt(20)))))))))))))))))))) 1.7579327566180045 = sqrt(1+sqrt(2+sqrt(3+...+sqrt(21)))))))))))))))))))))

Consider now the series of 's, defined (for

n= 0, 1, 2, 3, ...) as follows:

We can adapt TAU.C to give TAU2.C, which calculates the series of 's, and gives the following results:

Thus we see that:This program calculates the series of tau(n), where tau(n) = sqrt(1^n + sqrt(2^n + sqrt(3^n + sqrt(4^n + ..,)))) tau(0) = 1.6180339887498949 tau(1) = 1.7579327566180045 diff = 0.139899 tau(2) = 1.9426554227639874 diff = 0.184723 tau(3) = 2.1767885971334420 diff = 0.234133 tau(4) = 2.4674045317177931 diff = 0.290616 tau(5) = 2.8234815128342032 diff = 0.356077 tau(6) = 3.2559047054957255 diff = 0.432423 tau(7) = 3.7777293307595237 diff = 0.521825 tau(8) = 4.4045817066589290 diff = 0.626852 tau(9) = 5.1551576862917514 diff = 0.750576 tau(10) = 6.0518133295215266 diff = 0.896656 tau(11) = 7.1212594166022409 diff = 1.069446 tau(12) = 8.3953802511653048 diff = 1.274121 tau(13) = 9.9122027969110444 diff = 1.516823 tau(14) = 11.7170469518366320 diff = 1.804844

- Taking powers of 2, 3, etc., does not cause the series of partial sums to become infinite;

each(n)is a particular real number.- The series of sums, i.e. the series of
(n)'s, does not tend to a limit;

for any real numberrthere is somensuch that(n) > r.

Finally we may note from the case of

(0)that

which is 1.618033989..., also known as , the "golden mean" — yet another mathematical oddity. Actually we can prove this equality quite easily: , so , so solving this quadratic equation for

(0)gives .

The Greek Alphabet Mathematical Software Hermetic Systems Home Page