Pennsylvania State University MATH 141 CALC ANLY GEOM II In-Class Worksheet Week 10 Page. Solutions. Winter 2023

Math 141 In-Class Worksheet Week 10 Page 1 of ?? Question 1 For each of the following series, determine whether they converge or diverge. You must name the test(s) used, as well as show the details of the test(s). (a) 1 X n =1 21=n pn (b) 1 X n =1 3 + 2 cos n n2 . . (c) 1 X n =2 1 n(ln n)1=4 Math 141 In-Class Worksheet Week 10 Page 2 of ?? (d) 1 X n =1 3n2 + 5n + 2 pn6 + n2 + 1 Question 2 For each of the following series, determine whether they converge absolutely, converge conditionally, or diverge. You must name the test(s) used, as well as show the details of the test(s). (a) 1 X n =1 (-1)n n2 + 4 (b) 1 X n =2 (-1)n n (ln n)1=4 Math 141 In-Class Worksheet Week 10 Page 3 of ?? (c) 1 X n =1 (-1)n pn + 4 . Question 3 Later in the course we will discuss the fact that π = 1 X n =1 (-1)n-1 4 2n - 1 = 4 - 4 3 + 4 5 - 4 7 + 4 9 - · · · : (a) Find the minimum number of terms, N, such that the Alternating Series Error Bound guarantees jπ - SNj ≤ 10-3; where SN = NX n =1 (-1)n-1 4 2n - 1 is the Nth partial sum. Math 141 In-Class Worksheet Week 10 Page 5 of ?? (b) What is the actual error jπ - SNj for the N you obtained above? In Desmos you can type sum" and it should be clear how to set this up without needing to code a for loop from scratch. Question 4 (a) Show that if an ≥ 0 and 1 X n =1 an converges, that 1 X n =1 (an)2 does too. (b) Show that the converse of the above statement is false by providing a counterexample | that is, find a series with positive terms 1 X n =1 an such that 1 X n =1 (an)2 converges but 1 X n =1 an diverges. Hint: You can do this with an appropriate p-series.