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. 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:

THEOREM 1 Dichotomy for Positive Series Recall that an increasing sequence converges if it is bounded above. Otherwise, it diverges. 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.

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

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

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

The sum of the reciprocal powers n−p is called a p-series. THEOREM 3 Convergence of p-Series The infinite series converges if p > 1 and diverges otherwise.

Another powerful method for determining convergence of a positive series is comparison. Suppose that 0 ≤ an ≤ bn. THEOREM 4 Comparison Test Assume that there exists M > 0 such that 0 ≤ an ≤ bn for n ≤ M.

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

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

Using the Comparison Correctly Study the convergence of

Suppose we wish to study the convergence of We might compare S with But, unfortunately, the inequality goes in the wrong direction: So we cannot use the Comparison Test to say anything about our larger series. In this situation, the following variation of the Comparison Test can be used.

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

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