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Mathematics LibreTexts

8: Sequences and Series

 

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This chapter introduces sequences and series, important mathematical constructions that are useful when solving a large variety of mathematical problems. The content of this chapter is considerably different from the content of the chapters before it. While the material we learn here definitely falls under the scope of "calculus,'' we will make very little use of derivatives or integrals. Limits are extremely important, though, especially limits that involve infinity. 

One of the problems addressed by this chapter is this: suppose we know information about a function and its derivatives at a point, such as \(f(1) = 3\), \(f^\prime(1) = 1\), \(f^{\prime\prime}(1) = -2\), \(f^{\prime\prime\prime}(1) = 7\), and so on. What can I say about \(f(x)\) itself? Is there any reasonable approximation of the value of \(f(2)\)? The topic of Taylor Series addresses this problem, and allows us to make excellent approximations of functions when limited knowledge of the function is available.

  • 8.1: Sequences
    We commonly refer to a set of events that occur one after the other as a sequence of events. In mathematics, we use the word sequence to refer to an ordered set of numbers, i.e., a set of numbers that "occur one after the other.'' For instance, the numbers 2, 4, 6, 8, ..., form a sequence. The order is important.
  • 8.2: Infinite Series
    This section introduces us to series and defined a few special types of series whose convergence properties are well known: we know when a p-series or a geometric series converges or diverges. Most series that we encounter are not one of these types, but we are still interested in knowing whether or not they converge.
  • 8.3: Integral and Comparison Tests
    There are many important series whose convergence cannot be determined by these theorems, though, so we introduce a set of tests that allow us to handle a broad range of series including the Integral and Comparison Tests
  • 8.4: Ratio and Root Tests
    The comparison tests of the previous section determine convergence by comparing terms of a series to terms of another series whose convergence is known. This section introduces the Ratio and Root Tests, which determine convergence by analyzing the terms of a series to see if they approach 0 "fast enough.''
  • 8.5: Alternating Series and Absolute Convergence
    In this section we explore series whose summation includes negative terms. We start with a very specific form of series, where the terms of the summation alternate between being positive and negative.
  • 8.6: Power Series
    So far, our study of series has examined the question of "Is the sum of these infinite terms finite?,'' i.e., "Does the series converge?'' We now approach series from a different perspective: as a function. Given a value of x , we evaluate f(x) by finding the sum of a particular series that depends on x (assuming the series converges). We start this new approach to series with a definition.
  • 8.7: Taylor Polynomials
    A Taylor polynomial is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
  • 8.8: Taylor Series
    The difference between a Taylor polynomial and a Taylor series is the former is a polynomial, containing only a finite number of terms, whereas the latter is a series, a summation of an infinite set of terms.

Contributors

  • Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/