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5: Sequences and Series

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    • 5.1: Sequences
      In this section, we introduce sequences and define what it means for a sequence to converge or diverge. We show how to find limits of sequences that converge, often by using the properties of limits for functions discussed earlier. We close this section with the Monotone Convergence Theorem, a tool we can use to prove that certain types of sequences converge.
    • 5.2: Infinite Series
      In this section we define an infinite series and show how series are related to sequences. We also define what it means for a series to converge or diverge. We introduce one of the most important types of series: the geometric series. We will use geometric series in the next chapter to write certain functions as polynomials with an infinite number of terms. This process is important because it allows us to evaluate, differentiate, and integrate complicated functions by using polynomials.
    • 5.3: The Divergence and Integral Tests
      The convergence or divergence of several series is determined by explicitly calculating the limit of the sequence of partial sums. In practice, explicitly calculating this limit can be difficult or impossible. Several tests exist that allow us to determine convergence or divergence for many types of series.Here, we discuss two of these tests: the divergence test and the integral test. We will examine several other tests in the rest of this chapter and then summarize how and when to use them.
    • 5.4: Comparison Tests
      We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. Typically these tests are used to determine convergence of series that are similar to geometric series or p-series.
    • 5.5: Alternating Series
      In this section we introduce alternating series—those series whose terms alternate in sign. We will show in a later chapter that these series often arise when studying power series. After defining alternating series, we introduce the alternating series test to determine whether such a series converges.
    • 5.6: Ratio and Root Tests
      In this section, we prove the last two series convergence tests: the ratio test and the root test. These tests are nice because they do not require us to find a comparable series. The ratio test will be especially useful in the discussion of power series in the next chapter. Throughout this chapter, we have seen that no single convergence test works for all series. Therefore, at the end of this section we discuss a strategy for choosing which convergence test to use for a given series.
    • 5.7: Chapter 9 Review Exercises

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