EXPONENTIAL SERIES a^x REGION OF CONVERGENCE WITH a,x ALGEBRAIC OR TRANSCENDENTAL

The Present Theory Examines Exponential Series a^x Region of Convergence With x And a>0 Algebraic or Transcendental And The Multiplication of Irrational Numbers. It Disproves Beal's Conjecture Proves Fermat's Last Theorem Finds Pythagora's Theorem As An Exception To the Theory And Proves Beal And Fermat Triplets Cannot Exist

James Constant

math@coolissues.com

        Through application of the ratio test, the exponential series e^x with general term xⁿ/n! is widely believed to be absolutely convergent for all values of x.¹ The ratio test limit is /(n+1) which 0 when x is finite and 1 when x is infinite. The ratio test, therefore, succeeds only when x is finite but fails when x is infinite. The region of convergence of the exponential series e^x, obtained using the ratio test, is all finite numbers . 

        If x=1, the exponential series e has a general term 1/n! whose absolute convergence follows at once from the ratio test whose limit is 1/(n+1)0. Unlike exponential series e^x exponential series e has no x weakening use of the ratio test.

        Consider now the region of convergence of the exponential series e^x, obtained using the comparison test with exponential series e whichrequires all numbers x 1. Ifx is rational the region of convergence is x 1 and if x is irrational the region of convergence is x<1.

        Thus, we have two different regions of convergence for exponential series e^x obtained by the ratio and comparison tests. The ratio test, developed from comparing a series with the geometric series², is a weak test for obtaining the region of convergence of exponential series e^x. The comparison test is a stronger test for obtaining the region of convergence of exponential series e^x, by comparing it with exponential series e. The distinction is important because different proofs are obtained using the ratio and comparison tests.

        The proof that exponential series e^x is irrational, with x rational or irrational ³, is obtained following the proof that exponential series e is irrational ⁴, and using the comparison test with exponential series e.First, form

                     <1     x 1                             (1)

from which we can deduce that exponential series e^x, with x rational or irrational, is irrational. For if the contrary is true, that is if e^x=p/q, where p and q are integers, we can choose n larger than q so that n!p/q must be an integer. Next, second, multiply both sides of equation (1) by n! and set e^x=p/q

            n>q         <1    x 1           (2)

whose left hand side is an integer and whose right hand side indicates a number, rational when x is rational and irrational when x is irrational, plus a non vanishing fraction, which makes equation (2) impossible.

        The exponential series e^x with x algebraic rational or irrational is absolutely convergent and is, therefore, determinate and irrational for all values of x1 obtained using the comparison test. If instead one uses the ratio test, the same result is obtained but the region of convergence is all finite numbers .  Series e^x with x<0 converges conditionally and may, therefore, be indeterminate and irrational or indeterminable.⁵

        The exponential series e^x with x transcendental may converge absolutely or conditionally and may, therefore, be determinate and irrational or may be indeterminate and irrational or indeterminable. Consider

            0<c 1     0<c<2                  (3)

in which 0<c for logarithms, c1 and c<2 are obtained using the comparison test with series e and the ratio test, respectively. 

        By the comparison test, series (3) converges absolutely and is determinate and irrational when 0<c 1 (lnc 0), diverges and is indeterminable elsewhere; by the ratio test, series (3) converges conditionally and is indeterminate and irrational or indeterminable when 1<c<2 (lnc>0), and diverges and is indeterminable elsewhere. Accordingly, by the comparison test, when x=lnc<0 (0<c<1) exponential series e^x converges conditionally and is indeterminate and irrational or indeterminable, diverges and is indeterminable elsewhere; by the ratio test, when x=lnc>0 (1<c<2) exponential series e^x converges absolutely and is determinate and irrational, diverges and is indeterminable elsewhere.

        Consider now the general exponential series obtained with the comparison test with series e and the ratio test, respectively.

                0<|xlna| 1                     (4)

in which a is a finite number not a function of series running term n. The inequality represents the region of absolute and conditional convergence valid for all values of x,a determined by the comparison test with series e. Deleting   1 it represents the regions of absolute and conditional convergence valid for all finite values of x,a determined by the ratio test. What can we say about numbers y, a, and x? 

        To begin with, by the comparison test, when xnla>0 series (4) converges absolutely and is determinate and irrational for all values of x,a, when xnla<0 series (4) converges conditionally and is indeterminate and irrational or indeterminable for all values of x,a; by the ratio test, when xlna>0 series (4) converges absolutely and is determinate and irrational for all finite values of x,a, when xlna<0 series (4) converges conditionally and is indeterminate and irrational or indeterminable for all finite values of x,a.

        In what follows, since conditionally convergent series (3) and (4) are indeterminate or indeterminable, I refrain from manipulating terms to change sums in these type series. I classify the strength of different types of exponential series (4) as Highest strength xlna>0:  both x and lna are absolutely convergent; Medium strength xlna<0: x or lna is conditionally convergent; Low strength xlna<0 both x and lna are conditionally convergent. By strength, I mean absolute series do not but conditional series may leave open questions.  While these differences mean little in practical applications they are critical in theoretical investigations.

        I discuss six cases. In each case a>0or x is algebraic unless otherwise noted being transcendental.

I.        xlna=1     Say numbers like y=a^x with x transcendental

a=e^1/x=e^lna

series (4) converges absolutely when xlna=1

II.    xlna>0     Say numbers like y=a^x with

a>1 (lna>0), x>0           or           0<a<1 (lna<0),x<0

series (4) converges absolutely when 0<xlna<1  and diverges elsewhere.

Note 2: I treat series for lna absolutely convergent when lna<0 and conditionally convergent when lna>0.

III.    xlna<0     Say numbers like y=a^x with

0<a<1 (lna<0),x>0           or           a>1 (lna>0),x<0

series (4) converges conditionally when 1<xlna<0 and diverges elsewhere.

Note 3: See Note 2.

IV.    x and/or a is transcendental

        Transcendental numbers x and/or a can be incorporated in similar manner. For example, if x=lnc,y is a function of product  lnclna which exists only when c>0 and a>0. Numbers y,a,x are obtained following Cases I - III.

V.     Products x=ij with i,j irrational

series (4) converges absolutely when 0<ijlna<1 and diverges elsewhere; series (4)  converges conditionally when 1<ijlna<0 and diverges elsewhere.

 Note 4:  In general, products ij with at least one of i,j transcendental are irrational numbers. See http://www.coolissues.com/mathematics/productsij/irratnos.htm 

Note 5: In Case I, I assumed that product xlna=1 with x irrational. When determining the proximity of irrational product xlna to rational numbers, Hurwitz's irrational number theorem gives the best rational  approximation for an arbitrary irrational number xlna.⁶

VI.     Beal, Fermat, Pythagoras

            As an interesting application, consider Beal's proposed equality

            y=(uⁱ+v^j)^1/k     k>2     y,u,v.k,i,j integers                                 (5)

whose negation is Fermat's Last Theorem when k=i=j, and when k=i=j=2 is Pythagora's Theorem. In equation (5) a=uⁱ+v^j(lna>0) and 1/k>0 which are Case II numbers. Accordingly,

a>2 (lna>0), x=1/k>0

series (4) converges absolutely when 0<lna  k and converges conditionally when k<lna<0, thus disproving Beal's conjecture and proving Fermat's Last Theorem.

Note 6:  Since a>1 (lna>0), I treat series for lna conditionally determinate and irrational.

Note 7:  Pythagora's Theorem is an exception to the present theory because it admits Pythagorean triplets y,u,v of some rational numbers. Pythagorean triples say that series (4) does not exist for some finite values of y²=a, within its region(s) of  convergence.

Note 8:  Beal and Fermat triplets do not exist for otherwise Euclid's formulas for triplets can be written as 

u =(r² - s²)^2/a          v =  (2rs)^2/b         y = (r² + s²)^2/m ,         r,s integers 

which are impossible when a,b,m>2.

VII.    Euler's Function

           If Euler's function is written as a^x with a=1/n and compared with series e, one obtains series (4) with exceptions that a is no longer a finite number but  is a function of series (4) running term n and, second, lna<0. Formally

             0<|xlna|<1         a=1/n                                                              

but since a=1/n, as noo  the inequality fails both the comparison with series e and ratio tests.  

        Euler's function is the well known p series which converges absolutely when x>1 and diverges when x1. The region of convergence of Euler's function was determined independently, not using the comparison or ratio tests.⁷  The p series independent determination of convergence says nothing of running term n, which makes its region of absolute convergence x>1 valid for all values of x. 
       
        I conclude that the exponential series formulation of the p series offers no advantages for determining the convergence properties of series (4), a conclusion important in investigations involving Euler's function.
    
VIII.    Gelfund, Schneider

        Gelfund (1934) and Schneider (1934) first examined numbers of the form a^x.⁸ From the Gelfund and Schneider theorem (GST), such numbers are transcendental if x is algebraic and irrational and a is algebraic. The transcendence of the following numbers follows immediately from the GST: 2^1/2,. However, there are important differences between the two theories. In general, while the present theory proves transcendence based on the exponential series, the GST theory proves transcendence based on equivalences.⁹

²  R. Courant, Vol I pages 378-379

³Number e^x is believed to be irrational for rational x. http://mathworld.wolfram.com/IrrationalNumber.html

⁴ R. Courant, Vol I pages 336-337

⁵ R. Courant, Vol I pages 369-375, Conditionally convergent series are indeterminate and irrational or indeterminable

⁶ http://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html

⁷  R. Courant, Vol I pages 380-382;  Euler's function fails the ratio test.

⁸ http://mathworld.wolfram.com/GelfondsTheorem.html

⁹ For proof of GST see Hille, Einar. "Gelfond's Solution of Hilbert's Seventh Problem." The American Mathematical Monthly, Vol. 49, No. 10 (Dec., 1942), pp. 654-661; also a proof retrieved from Wikipedia http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes8.pdf

Copyright © 2009 James Constant

By same author: http://www.coolissues.com/mathematics/sameauthor.htm