MATH 1272 Lecture Notes - Lecture 1: Alternating Series Test, Alternating Series, Natural Number
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An alternating series is a series whose terms alternate between positive and negative. Suppose w have an alternating series that can be written as either. Has lim n(cid:736) of bn =0. Bn=1/n 0 for all n (n is a natural number) {bn} is decreasing since f(x) =1/x is decreasing on [1, ) Warning- this test dos not give us the sum (value) of the series. If we don"t meet the conditions, we cannot use the test. Let series an be a series and m 1 a natural number. Converges if and only if the tail sum. Which is the (tail of the) alternating harmonic series, which converges by earlier work. So the original series an converges so by the tail convergence theorem. *notice that in this limit l is always between sn and sn+1. By dominating terms, which we can use since its a limit as n (cid:736) . So alas, we cannot use alt series test.
