Final Thoughts on Infinite Sequences

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Recall our Learning Objectives

In this lesson we covered

  • the definition of convergence of an infinite sequence
  • various theorems on limits of infinite sequences, including
  • the theorem relating between the limit of a function and a limit of a sequence
  • the limit laws for infinite sequences
  • the squeeze theorem for infinite sequences
  • the theorem on the limit of the absolute value of a sequence

After reading this lesson and completing a suitable number of exercises, you should be able to, given an infinite series

  • determine whether an infinite sequence converges using the theorems presented in this lesson

What We Didn't Cover

In this lesson, we did not explore the concepts of

  • increasing and decreasing sequences
  • the Monotonic Sequence Theorem
  • bounded sequences

Students are not expected to be familiar with these concepts, but they are are needed to prove theorems that appear later in the ISM. Some of these concepts appear in the ISM Appendices, and we will refer to them when necessary.

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