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Published in:
A Brief Historical Introduction Read chapter Read first chapter

2015 | OriginalPaper | Chapter

6. Infinite Sequences III

Authors : Miklós Laczkovich, Vera T. Sós

Published in: Real Analysis

Publisher: Springer New York

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Abstract

In Theorem 4.10, we proved that for a sequence to converge, a necessary condition is the boundedness of the sequence, and in our example of the sequence (−1) n , we saw that boundedness is not a sufficient condition for convergence.

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previous chapter Infinite Sequences II
next chapter Rudiments of Infinite Series
Footnotes
1
This notation was introduced by Leonhard Euler (1707–1783), Swiss mathematician.
 
2
In practice, (6.4) is not a very useful approximation. If we wanted to approximate e to 10 decimal points, we would have to compute a 1010th power. We will later give a much faster approximation method.
 
3
In fact, one can also show that e is what is called a transcendental number, meaning that it is not a root of any nonzero polynomial with integer coefficients. We will prove irrationality in Exercises 12.87 and 15.23, while one can prove transcendence as an application of integration.
 
4
James Stirling (1692–1770) Scottish mathematician.
 
5
Bernhard Bolzano (1781–1848), Italian–German mathematician, and Karl Weierstrass (1815–1897), German mathematician.
 
6
Augustin Cauchy (1789–1857), French mathematician.
 
Literature
1.
Davidson, K.R., Dosig, A.P.: Real Analysis and Applications. Theory in Practice. Springer, New York (2010)
2.
Erdős, P., Surányi, J.: Topics in the Theory of Numbers. Springer, New York (2003)
3.
Euclid: The Thirteen Books of the Elements [Translated with introduction and commentary by Sir Thomas Heath]. Second Edition Unabridged. Dover, New York (1956)
4.
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1975)
5.
Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)
6.
Rademacher, H., Toeplitz, O.: Von Zahlen und Figuren. Springer, Berlin (1933) [English translation: The Enjoyment of Mathematics]. Dover, New York (1990)
7.
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)
8.
Zaidman, S.: Advanced Calculus. An Introduction to Mathematical Analysis. World Scientific, Singapore (1997)
Metadata
Title
Infinite Sequences III
Authors
Miklós Laczkovich
Vera T. Sós
Copyright Year
2015
Publisher
Springer New York
Book
Real Analysis

Print ISBN: 978-1-4939-2765-4

Electronic ISBN: 978-1-4939-2766-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-1-4939-2766-1_6

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