MATH 115 Midterm: MATH 115 UPenn 115Fall 08 Exams
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November 16, 2007: (15 points) find the sum of the series: Expressing the series in terms of a geometric series, we nd. Thus the series becomes a number plus a geometric series with a = 2 it converges and: = 3: (15 points) find the sum of the following series. 2 n + 2(cid:19) n + 1 . Since in the telescopic sum, all but these four summands cancel out. = lim n sn : (15 points) Let"s look at the n-th term of the sum: an = 2n ln(n+1) then: lim n an = lim n . This is a limit with inde nite form . Thus applying l"hospital"s rule we get: lim n an = lim n . So, the series diverges by the divergence test: (15 points) The series converges by the alternating seris test since it is alternating and the terms are decreasing to zero.
