convergence_tests - (12.7 Strategy for Convergence Tests To...

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(12.7) Strategy for Convergence Tests To test series for convergence, there are some rough rules of thumb to follow to decide which convergence tests to apply. Perhaps the first thing to consider for a series a n is whether lim n →∞ a n = 0. If that isn’t true, the series cannot converge. If the series is alternating, try the alternating series test. If a n has roots and powers of n in it (e.g., n n 3 +1 ), a comparison test with a p -series is probably in order. (In this case, compare with 1 n 5 / 2 .) If a n has exponents or factorials in it, the ratio test is probably the one to try. Occasionally, the root test is good to try, if a n consists of something raised to the n -th power, such as a n = 2 n 3 n +1 n . The integral test is not used very often. Occasionally, there will be a series that’s not amenable to the other tests, where the integral test works. If you have to determine if a series converges absolutely or conditionally, it is usually an alternating series. Look at the series with all terms positive (look at
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