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Calculus II pp 561-644 | Cite as

Infinite Series

  • Jerrold Marsden
  • Alan Weinstein
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

The decimal expansion \(\tfrac{1}{3} = 0.3333 \ldots\) is a representation of \(\tfrac{1}{3}\) as an infinite sum \(\tfrac{3}{{10}} + \tfrac{3}{{100}} + \tfrac{3}{{1000}} + \tfrac{3}{{10,000}} + \cdots\). In this chapter, we will see how to represent numbers as infinite sums and to represent functions of x by infinite sums whose terms are monomials in x. For example, we will see that
$$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$$
and
$$\sin x = x - \frac{{{{x}^{3}}}}{{1 \cdot 2 \cdot 3}} + \frac{{{{x}^{5}}}}{{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} - \cdots .$$

Keywords

General Solution Power Series Taylor Series Infinite Series Integral Test 
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.

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Copyright information

© Springer-Verlag New York Inc. 1985

Authors and Affiliations

  • Jerrold Marsden
    • 1
  • Alan Weinstein
    • 2
  1. 1.Control and Dynamical Systems 107–81California Institute of TechnologyPasadenaUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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