1.)

Determine whether the following series converges. E (-1 ) * + 1KK K = 1 . . . Let a 2 0 represent the magnitude of the terms of the given series. Select the correct choice below and fill in the answer box(es) to complete your choice. O A. The series diverges because ak = is nondecreasing in magnitude for k greater than some index N. O B. The series converges because ak = and for any index N, there are some values of k > N for which ak + 1 2 ak and some values of k > N for which ak + 1 sak. O C. The series diverges because ak = and for any index N, there are some values of k > N for which ak + 1 2 ax and some values of k > N for which ak + 1 Sak. O D. The series converges because ak = is nondecreasing in magnitude for k greater than some index N. O E. The series converges because a = is nonincreasing in magnitude for k greater than some index N and lim ak = K-+0o O F. The series diverges because ak = is nonincreasing in magnitude for k greater than some index N and lim ak = K-+00Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10 2. Although you do not need it. the exact value of the series is given. The number of terms that must be summed is D. (Simplify your answer.) Determine how manyr terms of the following convergent series must be summed to be sure that the remainder is less than 105 in magnitude. 3(1)k Z k! K=O The number of terms that must be summed is :l. (Simplify your answer.) Estimate the value of the following convergent series with an absolute error less than 10 3. m (Round to three decimal places as needed.) Determine whether the following series converges absolutely, converges conditionally, or diverges. co co k+1 K+1 ('1) 2(_1) ak= z 11 k=1 K21 k? Find lim ak. Select the correct choice below and, if necessary, ll in the answer box to kioo complete your choice. [:3 A. lim 3k = Kmo [:3 B. The limit does not exist. Determine whether the following series converges absolutely, converges conditionally, or diverges. 2 cos k E K6 K =1 Find lim a . Select the correct choice below and, if necessary, fill in the answer box to K-+00 complete your choice. O A. lim ak = K=00 O B. The limit does not exist.00 Does the series 2 (1)n Fl 7 n+2 converge absolutely, converge conditionally, or diverge? n = 1 Choose the correct answer below and, if necessary, ll in the answer box to complete your choice. {:1 A. The series converges conditionally per the Alternating Series Test and because the limit used in the Root Test is The series diverges because the limit used in the Ratio Test is not less than or equal to 1. The series converges conditionally per the Alternating Series Test and the 1 Comparison Test with 2 6. n=1rI 00 The series converges absolutely per the Comparison Test with 2 1 5' n=1 n The series converges absolutely because the limit used in the nth-Term Test is The series diverges because the limit used in the nth-Term Test is not zero. Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 103. (1)" 2 2 K=1 k The number of terms that must be summed is (Round up to the nearest integer as needed.)