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Mathematics LibreTexts

4: Convergence of Sequences and Series


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  • 4.1: Sequences of Real Numbers
    We can add two numbers together by the method we all learned in elementary school. Or three. Or any finite set of numbers, at least in principle. But infinitely many? What does that even mean? Before we can add infinitely many numbers together we must find a way to give meaning to the idea. To do this, we examine an infinite sum by thinking of it as a sequence of finite partial sums.
  • 4.2: The Limit as a Primary Tool
    The formal definition of the convergence of a sequence is meant to capture rigorously our intuitive understanding of convergence. However, the definition itself is an unwieldy tool. If only there was a way to be rigorous without having to run back to the definition each time. Fortunately, there is a way. If we can use the definition to prove some general rules about limits then we could use these rules whenever they applied and be assured that everything was still rigorous.
  • 4.3: Divergence of a Series
  • 4.E: Convergence of Sequences and Series (Exercises)

Thumbnail: Leonhard Euler. Image used with permission (Public Domain; Jakob Emanuel Handmann).


  • Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)