In this section, we consider positive series

3. The next three sections develop techniques for determining whether an infinite series converges or diverges. This is easier than finding the sum of an infinite series, which is possible only in special cases. In this section, we consider positive series where an > 0 for all n. We can visualize the terms of a positive series as rectangles of width 1 and height an (Figure 1). The partial sum is equal to the area of the first N rectangles. The key feature of positive series is that their partial sums form an increasing sequence: for all N. This is because SN+1 is obtained from SN by adding a positive number:

4. Recall that an increasing sequence converges if it is bounded above. Otherwise, it diverges (Theorem 6, Section 10.1). It follows that a positive series behaves in one of two ways (this is the dichotomy referred to in the next theorem). THEOREM 1 Dichotomy for Positive Series (i) The partial sums SN are bounded above. In this case, S converges. Or, (ii)The partial sums SN are not bounded above. In this case, S diverges. • Theorem 1 remains true if an ≥ 0. It is not necessary to assume that an > 0. • It also remains true if an > 0 for all n ≥ M for some M, because the convergence of a series is not affected by the first M terms.

5. Assumptions Matter The dichotomy does not hold for a nonpositiveseries. Consider The partial sums are bounded (because SN = 1 or 0), but S diverges.

6. Our first application of Theorem 1 is the following Integral Test. It is extremely useful because integrals are easier to evaluate than series in most cases. THEOREM 2 Integral Test Let an = f (n), where f (x) is positive, decreasing, and continuous for x ≥ 1.

7. diverges. The Harmonic Series Diverges Show that an = f (n), where f (x) is positive, decreasing, and continuous for x ≥ 1. THEOREM 2 Integral Test Let an = f (n), where f (x) is positive, decreasing, and continuous for x ≥ 1.

9. The sum of the reciprocal powers n−p is called a p-series. THEOREM 3 Convergence of p-SeriesThe infinite series converges if p > 1 and diverges otherwise. Here are two examples of p-series:

10. Another powerful method for determining convergence of positive series is comparison. Suppose that 0 ≤ an ≤ bn. Figure 4 suggests that if the larger sum converges, then the smaller sum also converges. Similarly, if the smaller sum diverges, then the larger sum also diverges. THEOREM 4 Comparison TestAssume that there exists M > 0 such that 0 ≤ an ≤ bn for n ≤ M.

11. THEOREM 2 Sum of a Geometric Series Let c 0. If |r| < 1, then THEOREM 4 Comparison TestAssume that there exists M > 0 such that 0 ≤ an ≤ bn for n ≤ M.

12. The divergence of (called the harmonic series) was known to the medieval scholar Nicole d’Oresme (1323–1382).

13. Using the Comparison Correctly Study the convergence of Fortunately, the Integral Test can be used. The substitution u = lnx yields

14. Suppose we wish to study the convergence of Thus we might try to compare S with Unfortunately, however, the inequality goes in the wrong direction: Although the smaller series converges, we cannot use the Comparison Theorem to say anything about our larger series. In this situation, the following variation of the Comparison Test can be used.

15. THEOREM 5 Limit Comparison Test Let {an} and {bn} be positive sequences. Assume that the following limit exists:

16. CONCEPTUAL INSIGHT To remember the different cases of the Limit Comparison Test, you can think of it this way. If L > 0, then an ≈ Lbn for large n. In other words, the series are roughly multiples of each other, so one converges if and only if the other converges. If L = ∞, then an is much larger than bn (for large n), so if Finally, if L = 0, then bn is much larger an and the convergence of

17. THEOREM 5 Limit Comparison Test Let {an} and {bn} be positive sequences. Assume that the following limit exists:

18. THEOREM 5 Limit Comparison Test Let {an} and {bn} be positive sequences. Assume that the following limit exists: