Consider the following convergent series. Complete parts a through d below. 3 Σ k = 1 a. Use an integral to find an upper bound for the remainder in terms of n. 3 The upper bound for the remainder is...

Extracted text: Consider the following convergent series. Complete parts a through d below. 3 Σ k = 1 a. Use an integral to find an upper bound for the remainder in terms of n. 3 The upper bound for the remainder is 4n 4 b. Find how many terms are needed to ensure that the remainder is less than 10-3. The minimum number of terms needed is 6. (Round up to the nearest whole number. Use the answer from part a to answer this part.) c. Use an integral to find lower and upper bounds (L, and U, respectively) on the exact value of the series. 3 3 and Un = Sn + 4n* Ln = Sn + %3D 4(n + 1)ª (Type expressions using n as the variable.) d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. Using ten terms of the series, the value lies in the interval (Do not round until the final answer. Then round to nine decimal places as needed. Use the answer from part c to answer this part.)


Extracted text: Consider the following convergent series. Then complete parts a through d below. 00 5 Σ 3k k= 1 a. Use an integral to find an upper bound for the remainder in terms of n. 5(3) -n In (3) The upper bound for the remainder is b. Find how many terms are needed to ensure that the remainder is less than 10-3 The minimum number of terms needed is 8. (Round up to the nearest whole number. Use the answer from part a to answer this part.) c. Use an integral to find lower and upper bounds (L, and Un, respectively) on the exact value of the series. Choose the correct answer below. O A. Ln = 2 5 5(3)-" 5 5(3) - (n + 1) Un = E + + In 3 3k k= 1 In 3 kz1 3k OB. 5(3) - (n+ 1) In 3 5(3) -k n Ln = 2 + In 3 Un = E 3-" 5(3) -(n + 1) + In 3 In 3 k= 1 k = 1 C. 5(3) - (n + 1) 5(3) -n Ln = 2 -Un = E - 3k k= 1 In 3 3k In 3 k = 1 d. Use the lower and upper bounds to find an interval in which the value of the series must lie if you approximate it using ten terms of the series. Using ten terms of the series, the value lies in the interval (OD (Use integers or decimals for any numbers in the expression. Do not round until the final answer. Then round to seven decimal places as needed. Use the answer from part c to answer this part.)