Problem #7: Determine the convergence or divergence of the series > n=1 Problem #7: (A) The series is absolutely convergent (B) There is not enough information to determine (C) The series is divergent (D) The series is conditionally convergent Select (-1)-¹(n+914) Save

7. Evaluate the series (-2)k+1 32k-1 k=2 8 8 2 3 3 (a) (b) 21 33 9 6 6 4 (e) (f) 7 11 (i) The series is convergent, but not equal to any of these answers. 8 45 (h) 4 (j) The series is divergent. (d)

Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent to 0, then the series > (-1)"+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 1 1 1 1 1 1 1 1 (-1)"+1bn, - 2 4 3 8. 4 16 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 4' 3'8'4'16’5'32' 1 1 } n 2n (a) Does this sequence {bn} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 (b) Show that this series diverges. (Hint: This series can also be written as Suppose that it did n 2n n=1 should converge. If you add a certain geometric series to it, you're adding two convergent series together, so you get another convergent series but do you?) (c) Is the Alternating Series Test wrong? Explain why not.

For problem 2 part 2: To get your comparison series for LCT, choose the series with leading term over leading term. So you want the series of sqrt{n} over n  "leading term over leading term" then go from there

1. If the series an is convergent and an > 0 for all n, then we also know that n=1 (a) Must be Convergent (b) Must be Divergent (c) Can be either Convergent or Divergent, depending on an M8 n=1 an n E

∞ 1. Let an be a POSITIVE infinite series (i.e. an > 0 for all n ≥ 1). Let ƒ be a continuous function n=1 with domain R. Is each of these statements true or false? If it is true, prove it. If it is false, prove it by providing a counterexample and justify that is satisfies the required conditions.

AQuestion Completion Status: Problem 1: Which of the following series converge by the alternating series test. Select all that apply. (-1)"n² 回》 Vn3 + 1 n=1 8 [ii] (-1)"-!el-n n=1 (-1)" [iii] ) 2+3n n=1 [iv] (-1)"n In(n) n=1 [i] [ii] [iv] Save All Answers Close Window Click Save and Submit to save and submit. Click Save All Answers to save all answers. pg.jpg MATH1132_Q11.2.pdf pg.jpg MATH1132_Q11.4.pdf MATH1132_Q11.3.pdf Essay 1 Makin....docx POF MacBook Pro