# How Gelfond find his limit for $\exp(\pi)$? [duplicate]

This question already has an answer here:

$$a_0 = \frac{1}{\sqrt 2}$$

$$a_{n+1} = \frac{( \sqrt {1 - a_n^2} -1)^2}{a_n^2}$$

$$\lim_{n \to \infty} \frac{4^{\frac{1}{2^n}}}{a_{n+1}^{\frac{1}{2^n}}} = \exp(\pi)$$

How did Gelfond find this nice result ? And how to prove it ? The 4 is trivial , but the rest is not. Is this related to trigonometry ? Is this related to continued fractions ?

Are there analogues known for cube roots ?

Notice a proof alone might not explain how he found the result.

## marked as duplicate by Paramanand Singh limits StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 30 '18 at 3:51

• mick Please check my edits, fairly minor, but primarily check those for the for $a_{n+1}$. It seemed as though you intended for the square root to extend over all of $1 -a_n^2$. – Namaste Jun 29 '18 at 11:31
• Yes I did amwhy. – mick Jun 29 '18 at 11:33
• The theory behind the result is difficult and you may get a brief explanation from my answer to the original question. The details would require some study of elliptic integrals and theta functions which are covered nicely in my blog. – Paramanand Singh Jun 30 '18 at 4:09
• Also Gelfond did not find this result. Rather the constant $e^\pi$ goes by the name Gelfond's constant. – Paramanand Singh Jun 30 '18 at 4:33
• A similar iteration can be given using same approach involving cube roots (or nth roots in general) but the iteration would be clumsy and hard to follow. – Paramanand Singh Jun 30 '18 at 8:38

This is the Gauss-Legendre algorithm non homogeneous version for the elliptic modulus k. That is, let $\, a_n = k(q_n) = (\theta_2(q_n)/\theta_3(q_n))^2 \,$ where $\, q_n = \exp(-\pi 2^n). \,$ We have $\, a_{n+1} = 4 q_n + O(q_n^3). \,$
Legendre used the following: $\, k_{n+1} = (1 - k'_n)/(1 + k'_n) = k_n^2/(1 + k'_n)^2 = (1 - k'_n)^2/k_n^2 \,$ where $\, 1 = k_n^2 + k_n'^2. \,$ The AGM of Gauss is $\, a_{n+1} \!=\! \frac12(a_n \!+\! b_n),$ $b_{n+1} \!=\! \sqrt{a_n b_n}, \,$ $c_{n+1} \!=\! \frac12(a_n\!-\!b_n). \,$ The connection is $\, k_n = c_n/a_n, \, k_n' = b_n/a_n. \,$ For more details, you can read the standard book Pi and the AGM by Borwein and Borwein.
By the way, the recursion as written $\, a_{n+1} = (\sqrt {1 - a_n^2} -1)^2/a_n^2, \,$ is not good from a numerical point of view. The subtraction of two numbers nearly equal causes drastic loss of significance. The equivalent recursion $\, a_{n+1} = a_n^2 / (1 + \sqrt {1 - a_n^2})^2 \,$ does not suffer from significance loss.