(BC only) Infinite series and tests for convergence

Learning objectives

Define an infinite series as the limit of its sequence of partial sums, so that it converges exactly when that limit exists, and apply this to a geometric series, which converges when its common ratio satisfies |r| < 1 and then sums to a/(1 − r) where a is the first term

Apply the nth term test for divergence, concluding divergence when the terms do not approach zero, and state explicitly that a limit of zero on its own proves nothing

Apply the integral test to a series whose terms come from a positive, continuous and decreasing function, and use it to establish that a p-series converges when p > 1 and diverges when p ≤ 1, the harmonic series being the divergent case p = 1

Apply the direct comparison test and the limit comparison test, choosing a geometric series or p-series for comparison and checking that the terms are positive

Apply the alternating series test by verifying that the terms decrease in magnitude and tend to zero, and apply the ratio test, which gives absolute convergence when the limit of the ratio of consecutive terms in magnitude is less than 1, divergence when it is greater than 1, and no conclusion when it equals 1

Distinguish absolute convergence, where the series of absolute values converges, from conditional convergence, where the series converges but the series of absolute values does not, and classify a given alternating series accordingly