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Unformatted text preview: 1 Alternating Series Last time we talked about convergence of series. Our two most important tests, the integral test and the (limit) comparison test, both required that the terms of the series were (eventually) positive. There is an important role played by series with alternating terms. It is, in a sense, much easier for series with terms of alternating terms to converge since there is cancellation between the terms of the series. Lets begin with a numerical example. Id like to consider two series S = k =1 1 k S = k =1 (- 1) k +1 k As always we let S N ,S N denote the partial sums of the series. Lets look at some partial sums: N S N S N 1 1 1 10 2.92 0.646 21 3.65 0.716 32 4.06 0.678 43 4.35 0.704 54 4.58 0.684 65 4.76 0.701 The partial sums of the harmonic series are not obviously converging. In fact we know that the harmonic series does not converge. The partial sums of the alternating harmonic series look like they may be converging. The hover around . 7. Id also like to note two facts, which will prove important later: The odd partial sums form a decreasing sequence while the even partial sums form an increasing sequence. Further the largest term is S 1 , and every odd term is larger than every even term. These facts will be true in general, and will let prove that (under certain conditions) alternating series generally converge.prove that (under certain conditions) alternating series generally converge....
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