请问这个如何做？ 第1页

The final result converges to the Gelfond constant .This is not an ordinary question.This sequence is mentioned in the book Mathematics by Experiment: Plausible Reasoning in the 21st Century.

The solution of Paramanand Singh is given below.

Let be the elliptic modulusand then we define complementary modulus .Further the complete elliptic integrals of the first kind are defined by

By the way the use of prime here does not denote derivative.

Starting with a modulus we define a new modulus via ascending Landen transformation

where is complementary to ie . It can be proved with some effort that

Using a little amount of algebra it can be seen that the relation between and is same as the relation between and and hence from the above equation we get

and therefore from the last two equations we get

Corresponding to elliptic modulus we have another variable called nome defined by

and it is possible to get an expression for in terms of via the use of theta functions. We have

where

The sequence in question is descending Landen sequence of elliptic moduli and using equation we have

The corresponding nomes have a relation between them as Thus we can see that

Next also note that as and using we have the relation

The desired relation follows easily from and and the relation