Pennsylvania State University MATH 141 CALC ANLY GEOM II Written Homework 9 Solutions. Winter 2023.

Math 141 Written Homework 9 Page 1 of 5 Question 1 Do the following series converge or diverge? Find the sum of any convergent series. For (a) and (b), you can assume the suggested pattern continues. (a) 1 - 1 2 + 1 4 - 1 8 + · · · (b) 1.1 + 1.12 + 1.13 + 1.14 + · · · (c) ∞ Xn =3 5(0.4n) (d) ∞ Xn =1 (-1)n 32n+2 Math 141 Written Homework 9 Page 2 of 5 Question 2 A patient takes d mg of medicine every 8 hours. Every 8 hours, 80% of the medicine in the patient’s bloodstream is eliminated. The medicine has side effects if there is ever 15 mg or more in the patient’s blood, but there needs to be at least 2 mg in the patient’s blood for it to be effective. (a) What is the maximum safe value for d? (Hint: Consider An, the amount of medicine in the patient’s blood immediately after taking the nth dose, and think about what must be true about An to ensure a safe dosing schedule. Write out very carefully what A1, A2, A3 are before trying to find a pattern for An.) (b) What is the minimum effective value for d? (Hint: Consider Bn, the amount of medicine in the patient’s blood immediately before taking the nth dose.) Math 141 Written Homework 9 Page 4 of 5 Question 3 Determine whether the series ∞ Xn =2 1 n(ln n)3/2 converges or diverges. Be sure to name any test(s) used as well as the key details. Question 4 Determine whether the series ∞ Xn =3 1 n · ln n · ln(ln n) converges or diverges. Be sure to name any test(s) used as well as the key details.