第九次作业(A).doc

第九次作業(A)12.84. If , then . By the Ratio Test, the series converges when , so When , the series diverges because it is the harmonic series; when , it is the alternating harmonic series, which converges by the Alternating Series Test. Thus, 9. . Now , so by the Ratio Test, When , we get the divergent series , and when , we get the divergent series . Thus, 20. , so by the Ratio Test, the series converges when . When , the series is , the convergent alternating harmonic series. When , the series becomes the divergent harmonic series. Thus, 30. We are given that the power series is convergent for and divergent when So by Theorem 3 it converges for at least and diverges for at least and Therefore: (a) It converges when ; that is, is convergent. (b) It diverges when ; that is, is divergent. (c) It converges when ; that is, is convergent. (d) It diverges when ; that is, is divergent. 12.94. with so and Note that converges converges, so the appropriate condition (from equation (1) is 9. = = = = = = The geometric series converges when so and 13. (a) from Exercise 3 from Theorem 2(i) with In the last step, note that we decreased the initial value of the summation variable by and then increased each occurrence of in the term by also note that . (b) from part (a) with (c) from part (b) To write the power series with rather than we will decrease each occurrence of in the term by and increase the initial value of the summation