For each of the series below select the letter from a to c that best applies and the letter from d to k that best applies. A possible answer is af, for example. A. The series is absolutely convergent. B. The series converges, but not absolutely. C. The series diverges. D. The alternating series test shows the series converges. E. The series is a p-series. F. The series is a geometric series. G. We can decide whether this series converges by comparison with a p series. H. We can decide whether this series converges by comparison with a geometric series. I. Partial sums of the series telescope. J. The terms of the series do not have limit zero. K. None of the above reasons applies to the convergence or divergence of the series. .1. i: c0::7r) 0 1 2' \"Z: nlog(6 l n) _ (2n + 6)! '3' Z W n 5' Z cosig'rrm) 0 1 .6- Z W 71:1 Here are the best answers for each of the series you provided: Series 1: Cos( NT) A. The series is absolutely convergent. E. The series is a p-series. Explanation: The p-series test states that if p _> 1, then the series ), - is convergent. The series cos ( NT) is absolutely convergent because it's terms are all bounded by 1 in absolute value. Series 2: 2\\mlog(6-712) B. The series converges, but not absolutely.G. we can decide whether this series converges by comparison with a p-series. Explanation: We can use the comparison test to compare this series to the p-series _, - 90 1. Since 3/2 > 1, the p-series converges, and so the given series must also converge. However, the terms of the given series do not approach zero, so it does not converge absolutely Series 3: 2 4 2146)( C. The series diverges. J. The partial sums of the series telescope. Explanation: The partial sums of the given series telescope to . As n approaches infinity, the term in parentheses approaches 0, so the series diverges. As Series 4: En 1 VnA. The series is absolutely convergent. G. We can decide whether this series converges by comparison with a geometric series. Explanation: We can use the comparison test to compare this series to the geometric series _, . Since n? > (1 + (n) > n, the geometric series converges, and so the given series must also converge. However, the terms of the given series do not approach zero, so it does not converge absolutely. Series 5: _ | cos. (NT) A. The series is absolutely convergent 1. Partial sums of the series telescope. Explanation: The partial sums of the given series telescope to (1 + cos(2NT) + cos(4NT) + ... + cos (2nNT ) ). As n approaches infinity, the term in parentheses approaches 5, so the series converges. Also, all of the terms of the series are bounded by 1, so the series is shenlutely commentabsolutely convergent. Series 6- log () A. The series is absolutely convergent. E. The series is a p-series Explanation: The p-series test states that if p # 1, then the series Will ! -; is convergent. The series -IS absolutely convergent because its terms are all bounded by its in absolute value, and Use, is ap- series with p I hope this helps