EDB β€” 20V

↑ ← β†’ ↓ view in whole PDF view in whole HTML

View

English

Exercise 8

[20V] Prerequisites:[20T].Let 𝛼>0,π›Όβˆˆβ„ be fixed. We know that, for every natural nβ‰₯1, there exists an unique 𝛽>0 such that 𝛽n=𝛼, and 𝛽 is denoted by the notation 𝛼n. (See e.g. Proposition 2.6.6 Chap. 2 Sec. 6 of the course notes [ 3 ] or Theorem 1.21 in [ 25 ] ). Given qβˆˆβ„š, we write q=n/m with n,mβˆˆβ„€,mβ‰₯1, we define

𝛼q=.𝛼nm.

Show that this definition does not depend on the choice of representation q=n/m; that

𝛼q=(𝛼m)n;

that for p,qβˆˆβ„š

𝛼q𝛼p=𝛼p+q,(𝛼p)q=𝛼(pq);

show that when 𝛼>1 then p↦𝛼p is strictly monotonic increasing.

Download PDF
Bibliography
  • [3] L. Ambrosio, C. Mantegazza, and F. Ricci. Complementi di matematica. Scuola Normale Superiore, 2021. ISBN 9788876426933. URL https://books.google.it/books?id=1QR0zgEACAAJ.
  • [26] Walter Rudin. Principles of Mathematical Analysis. McGraw–Hill, New York, 3rd edition, 1964.

Book index
  • real numbers
Managing blob in: Multiple languages
This content is available in: Italian English