A Transcendental Sequence and a Problem on Commutativity of Exponentiation of Real Numbers
| Autor: | Fouad Nakhli |
|---|---|
| Rok vydání: | 1989 |
| Předmět: | |
| Zdroj: | Mathematics Magazine. 62:185-190 |
| ISSN: | 1930-0980 0025-570X |
| DOI: | 10.1080/0025570x.1989.11977435 |
| Popis: | where the three dots indicate that there is an infinite number of 2's extending to the inside in the same manner. My next step was to explore experimentally the subsequences u2n-1 and u12n of the sequence u1 =2, un+1=2 1/ut. I found that these converge to the same limit which is the unique root of the equation xx = 2 (x = 1.5596104695). So it turned out that my "number" was really a number! (I was lucky that I chose 2 for the doodle. Had I chosen a different number, it could have been a different matter, as we shall see later). At this stage, the general sequence an was ready for attack. In what follows we study its behavior as m > 0 varies and we find its limit when it converges. During the course of this study, a solution is found spontaneously to the following curious problem: Given a positive real number m, how many ordered pairs (x, y) of positive real numbers exist such that |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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