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Alternating Series Test
ALTERNATING SERIES TEST Assume that (i) is an alternating series. (ii) The terms a_{n} are decreasing, a_{1} > a_{2} > ... > a_{n} > .... (iii) The terms approach zero, lim_{n→∞} a_{n} = 0. Then the series converges to a sum = S. Moreover, the sum S is between any two consecutive partial sums, S_{2n}< S < S _{2n+1}. Discussion We see from the graph in Figure 9.5.1 that the partial sums S_{n} alternately increase and decrease, but the change is less each time. The value of S_{n }"vibrates" back and forth and the vibration damps down around the limit S. PROOF The sequence of even partial sums is increasing. S_{2} < 5_{4} < ... < S_{2n} < ..., because S_{4} = S_{2} + (a_{3}  a_{4}), S_{6} = S_{4} + (a_{5}  a_{6}), etc. The sequence of odd partial sums is decreasing, S_{1} > S_{3} > S_{5} > ..., for S_{3} = S_{1}  (a_{2}  a_{3}), S_{5} = S_{3}  (a_{4}  a_{5}), etc. Figure 9.5.1 It follows that each even partial sum is less than S_{x}, S_{1} > S_{1}  a_{2} = S_{2}, S_{1} > S_{3}  a_{4} = S_{4}, S_{1} > S_{5}  a_{6} = S_{6}, etc. Theorem 1 (Section 9.4) shows that the increasing sequence of even partial sums converges, lim_{n→∞} S_{2n} = S. Given any infinite H, a_{2H+l} ≈ 0 and S_{2H} ≈ S, so S_{2H+1} = S_{2H} + a_{2H+1} ≈ S Therefore the sequence of all partial sums converges to S, and Finally, since the even partial sums are increasing and the odd partial sums are decreasing, we have the estimate S_{2n} < S < S_{2n+1}. Figure 9.5.2 shows a graph of the partial sums. Figure 9.5.2
The Cauchy Test for Divergence in Section 9.2 shows that if the terms an do not converge to zero the series diverges. We have now built up quite a long list of convergence tests. The next section contains one more important test, the Ratio Test. At the end of that section is a summary of all the convergence tests with hints on when to use them.


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