《数学通报》2014年1月号问题2162及其证明

作者：sqing55

              

                              

                       

 Expressions of the form

(1)

are called nested radicals. Herschfeld (1935) proved that a nested radical of real nonnegative terms converges iff  is bounded. He also extended this result to arbitrary   powers (which include continued square roots andcontinued fractions as well), a result is known as Herschfeld's convergence theorem.

Nested radicals appear in the computation of pi,

(2)

(Vieta 1593; Wells 1986, p. 50; Beckmann 1989, p. 95), in  trigonometrical values of cosine and sine for argumentsof the form , e.g.,

			(3)
			(4)
			(5)
			(6)

Nest radicals also appear in the computation of the  golden ratio

(7)

and plastic constant

(8)

Both of these are special cases of

(9)

which can be exponentiated to give

(10)

so solutions are

(11)

The  silver constant is related to the nested radical expression

(12)

There are a number of general formula for nested radicals (Wong and McGuffin). For example,

(13)

which gives as special cases

(14)

(, , ),

(15)

(), and

(16)

(). Equation (13) also gives rise to

(17)

which gives the special case for , , , and ,

(18)

Equation (◇) can be generalized to

(19)

for integers , which follows from

			(20)
			(21)
			(22)
			(23)
			(24)

In particular, taking  gives

(25)

(J. R. Fielding, pers. comm., Oct. 8, 2002).

Ramanujan discovered

(26)

which gives the special cases

(27)

for ,  (Ramanujan 1911; Ramanujan 2000, p. 323; Pickover 2002, p. 310), and

(28)

for , , and . The justification of this process both in general and in the particular example of , where  is  Somos's quadratic recurrence constant in given by Vijayaraghavan (in Ramanujan 2000, p. 348).

For a nested radical  of the form

(29)

to be equal a given  real number , it must be true that

(30)

so

(31)

and

(32)

An amusing nested radical follows rewriting the series for e

(33)

as

(34)

so

(35)

(J. R. Fielding, pers. comm., May 15, 2002).

                                

问题1119：《数学通报》2014年第1期问题2162题注记  anzhenping

 

一个漂亮等式的证明：倪伟sjz

                     《数学通报》2014年1、2月问题

《数学通报》问题2162的解答及推广  最帅数学