Calculus II pp 561-644 | Cite as

Infinite Series

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


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
$$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$$
$$\sin x = x - \frac{{{{x}^{3}}}}{{1 \cdot 2 \cdot 3}} + \frac{{{{x}^{5}}}}{{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} - \cdots .$$


General Solution Power Series Taylor Series Infinite Series Integral Test 
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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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