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Series
The concept of a series is defined and various properties of series are examined. Functional series (and, in particular, power series and related...
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Eisenstein Series
In this chapter we survey the theory of Eisenstein series. Our main goal is to state Langlands’ decomposition of...
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Power Series
We apply the ideas of the previous chapter to consider the convergence behavior of power series in general. We then discuss the fundamental results...
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Fourier Series
Chapter 1 covers the classical theory of Fourier series of...
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Fourier Series
In this chapter we consider the representations of functions in the form of series in trigonometric functions. This approach will be used for solving...
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Numerical Series and Series of Functions
Let V be a normed vector space. Given a sequence (a k)k in V , the associated “series” is the sequence (s n)n defined by...
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Sums and Series
In this chapter, we experience the problems of the second major area of the book, which is Sums and Series. Like in the first chapter, the reader...
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Some Important Series
We consider a few important series related to the Harmonic series and then Euler’s constant.
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Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling
B-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was...
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Gevrey Formal Power Series
As it is already the case when working with ordinary differential equations, it appears soon that the Poincaré asymptotics is not sufficient to study...
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Complex Power Series
The main goal of this chapter is to show that analytic functions can be represented as infinite power series. The key to proving this theorem is the...
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Embedding World Series
During December 1991–January 1992, Paul Erdős and I were working on the book Problems of pgom Erdős at my home in Colorado Springs. Ron Graham called...
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Series
With the help of series we will explain important functions. But this is future music, more about this in Chap. 24...
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Series
Now that we have developed a reasonable theory of limits of sequences, we will use that theory to develop a theory of infinite series...
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Series
Series are just a special type of sequences. The main feature of numerical series is that they lead us to finding convergence theorems which do not...
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Taylor and Maclaurin Series
In Example 12.1.9 , we have seen the power series...
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Change Point Analysis for Time Series
This volume provides a comprehensive survey that covers various modern methods used for detecting and estimating change points in time series and...
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Power Series
Power series are series in an Indefinite x. For some values of x the power series may converge, for others it may diverge. The range of all those x...
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Parameter Changes in Time Series Models
We develop in this chapter the asymptotic theory surrounding change point methods for many popular time series models. Although up to this point we...