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Sequences and Series of Functions

  • Matthew A. Pons
Chapter

Abstract

Sequences and series of functions are natural extensions of the topics discussed in Chaps. 2 and 3 and many of their properties are derived in a straightforward manner from the results covered there. The investigation of series of functions in the late eighteenth and early nineteenth centuries uncovered problems with the foundations of calculus which led to the restructuring of the subject and the birth of analysis as we know it today. Here we investigate the basic forms of convergence for sequences and series of functions with specific attention payed to the representation of functions as power series. This is propagated by our brief study of Taylor polynomials in Chap. 5 and we also provide a construction of a function which is continuous on \(\mathbb{R}\) but is not differentiable at any point.

Keywords

Power Series Taylor Series Differentiable Function Uniform Convergence Pointwise Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abbott, S.: Understanding Analysis. Springer, New York (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Axler, S.: Linear Algebra Done Right. Springer, New York (1996)Google Scholar
  3. 3.
    Axler, S.: Down with determinants! Am. Math. Mon. 102, 139–154 (1995)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bartle, R.G.: Return to the Riemann integral. Am. Math. Mon. 103, 625–632 (1996)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Beanland, K., Roberts, J.W., Stevenson, C.: Modifications of Thomae’s function and differentiability. Am. Math. Mon. 116(6), 531–535 (2009)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland, Amsterdam (1988)MATHGoogle Scholar
  7. 7.
    Bressoud, D.: A Radical Approach to Lebesgue’s Theory of Integration. MAA Textbooks. Cambridge University Press, Cambridge (2008)MATHGoogle Scholar
  8. 8.
    Bressoud, D.: A Radical Approach to Real Analysis, 2nd edn. Mathematical Association of America, Washington (2007)MATHGoogle Scholar
  9. 9.
    de Silva, N.: A concise, elementary proof of Arzelà’s bounded convergence theorem. Am. Math. Mon. 117(10), 918–920 (2010)MATHGoogle Scholar
  10. 10.
    Darst, R.B.: Some Cantor sets and Cantor functions. Math. Mag. 45, 2–7 (1972)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Drago, G., Lamberti, P.D., Toni, P.: A “bouquet” of discontinuous functions for beginners of mathematical analysis. Am. Math. Mon. 118(9), 799–811 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Halmos, P.: A Hilbert Space Problem Book. Springer, New York (1982)CrossRefMATHGoogle Scholar
  13. 13.
    Harier, E., Wanner, G.: Analysis by Its History. Springer, New York (1996)Google Scholar
  14. 14.
    Kirkwood, J.R.: An Introduction to Analysis. Waveland, Prospect Heights (1995)Google Scholar
  15. 15.
    Knopp, K.: Theory and Application of Infinite Series. Blackie, London (1951)MATHGoogle Scholar
  16. 16.
    Kubrusly, C.: Elements of Operator Theory. Birkhäuser, Boston (2001)CrossRefMATHGoogle Scholar
  17. 17.
    Lebesgue, H.: Measure and the Integral. Holden-Day, San Francisco (1966)MATHGoogle Scholar
  18. 18.
    MacCluer, B.D.: Elementary Functional Analysis. Springer, New York (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Martinez-Avendano, R., Rosenthal, P.: An Introduction to Operators on the Hardy-Hilbert Space. Springer, New York (2006)Google Scholar
  20. 20.
    Mihaila, I.: The rationals of the Cantor set. Coll. Math. J. 35(4), 251–255 (2004)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Nadler, S.B., Jr.: A proof of Darboux’s theorem. Am. Math. Mon. 117(2), 174–175 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Niven, I.: Numbers: Rational and Irrational. New Mathematical Library, vol. 1. Random House, New York (1961)Google Scholar
  23. 23.
    Niven, I.: Irrational Numbers. The Carus Mathematical Monographs, vol. 11. Mathematical Association of America, New York (1956)Google Scholar
  24. 24.
    Olmsted, J.M.H.: Real Variables. Appleton-Century-Crofts, New York (1959)MATHGoogle Scholar
  25. 25.
    Patty, C.W.: Foundations of Topology. Waveland, Prospect Heights (1997)Google Scholar
  26. 26.
    Pérez, D., Quintana, Y.: A survey on the Weierstrass approximation theorem (English, Spanish summary). Divulg. Mat. 16(1), 231–247 (2008)MATHMathSciNetGoogle Scholar
  27. 27.
    Radjavi, H., Rosenthal, P.: The invariant subspace problem. Math. Intell. 4(1), 33–37 (1982)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Rynne, B.P., Youngson, M.A.: Linear Functional Analysis. Springer, London (2008)CrossRefMATHGoogle Scholar
  29. 29.
    Saxe, K.: Beginning Functional Analysis. Springer, New York (2002)CrossRefMATHGoogle Scholar
  30. 30.
    Strichartz, R.S.: The Way of Analysis. Jones and Bartlett, Boston (1995)MATHGoogle Scholar
  31. 31.
    Thompson, B.: Monotone convergence theorem for the Riemann integral. Am. Math. Mon. 117(6), 547–550 (2010)CrossRefGoogle Scholar
  32. 32.
    Tucker, T.: Rethinking rigor in calculus: the role of the mean value theorem. Am. Math. Mon. 104(3), 231–240 (1997)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Matthew A. Pons
    • 1
  1. 1.North Central CollegeNapervilleUSA

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