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 which requires 0<x 1. Ifx is rational the region of convergence is 0<x 1 and if x is irrational the region of convergence is 0<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     0<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    0<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 values of 0<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 which allows negative as well as positive numbers x in equations (1) and (2). Series e^x with x<0 converges conditionally and may, therefore, be indeterminate and irrational or indeterminable.⁵ Series e^x with x<0 is, therefore, indeterminate and irrational or indeterminable.

        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. Series (3) converges absolutely and is determinate and irrational when 0<c 1 (lnc 0), converges conditionally and is indeterminate and irrational or indeterminable when 1<c<2 (lnc>0), and diverges and is indeterminable when c>2. Accordingly, the exponential series e^x with transcendental x=lnc<0 (0<c<1), is indeterminate and irrational or indeterminable, with transcendental x=lnc>0 (1<c<2) is determinate and irrational,and with transcendental x=lnc>0 (c>2) is indeterminable.

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

            0<xlna  1    <         (4)

which converges absolutely and is, therefore, determinate and irrational when xlna>0 and converges conditionally and is, therefore, indeterminate and irrational or indeterminable when xlna<0. In series (4), the inequalities represent the regions of convergence valid for all and for finite values of x,a respectively. What can we say about numbers y, a, and x?

        To begin with, a>0 for logarithms and transcendental lna converges absolutely and is, therefore, determinate and irrational when 0<a 1 (lna 0) and converges conditionally and is, therefore, indeterminate and irrational or indeterminable when a>1 (lna>0).

        In what follows, I treat series (4) absolutely determinate and irrational when 0<xlna  1 and conditionally determinate and irrational when all finite xlna>1 and conditionally determinate and irrational when all finite xlna<0. Likewise, I treat series (3) absolutely determinate and irrational when c 1 (lnc 0) and conditionally determinate and irrational when c>1 (lna>0). Such treatments are based on the choice of not rearranging series terms to obtain indeterminate and indeterminable results

        I discuss six cases. In each case a>0 or 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

are absolute values i.e., 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

are absolute values. i.e., series (4) converges absolutely when 0<xlna<1  and are conditional values. i.e., series (4) converges    conditionally when xlna>1

Note 2: I treat series for lna conditionally determinate and irrational when a>1 (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

are conditional values. i.e., series (4) converges conditionally when xlna<0

Note 3: I treat series for lna conditionally determinate and irrational when a>1 (lna>0).

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 

         For example, if i>0 and j>0,series (4) with a>0 (lna>0) and x=ij>0 is absolutely (0<ijlna  1) or conditionally (ijlna>1)  convergent and, if lna=1 (a=e) (0<ij 1) and (ij>1) respectively. 

 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 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

are absolute values. i.e., series (4) converges absolutely when 0<lna  k and are conditional values. i.e., series (4) converges conditionally when lna>k, 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.

        Gelfund (1934) and Schneider (1934) first examined numbers of the form a^x.⁷ 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 equivalences⁸

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 or indeterminable

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

⁷ http://mathworld.wolfram.com/GelfondsTheorem.html

⁸For 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

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