# 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.

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

### Contributors

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