![]() We know we can open it without fear of a sandwich explosion, but it will smell terrible.Ī partial sum S n is just an approximation of L. Here, L stands for limburger, which is the type of stinky cheese in our Pandora's box. Suppose we know, now, that we have an alternating series Σ a n that converges, and it converges to value L. Since all conditions are met, the AST says that the series converges. ![]() Sample ProblemĬan we use the AST to conclude that the series If the AST doesn't tell us that the series converges, we need to use another test, which might include the divergence test. We can only use the AST to conclude that a series converges. The catch: We can't use the AST to conclude a series diverges. If the terms are getting smaller and approaching zero, the partial sums will get closer to the line and so the series will converge.Īlternating Series Test (AST): If Σ a n is an alternating series, and ifįor all n (that is, the terms have strictly decreasing magnitude), and if Since the terms of an alternating series change sign, the partial sums for any alternating series will jump back and forth over some line. Line jumping is the idea behind our first convergence test, the alternating series test. We can find a formula for the terms of an alternating series the same way we can find a formula for the terms of an alternating sequence. Is an alternating series because the signs of the terms switch back and forth between positive and negative. The alternating series test can tell us if it's safe to open that box. This is for a Pandora's box full of American and Swiss grilled cheese sandwiches. The first tool in our arsenal of convergence tests is for alternating series, which is a series whose terms alternate in sign.
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