this is the requirement form

Writing: Solutions should be presented in a balanced form: combining words and sen tences which explain the line of reasoning, and also precise mathematical expressions, formulas and references justifying the steps you are taking are correct. In general, you must give a precise reason for why a sequence is divergent or convergent. Similarly, if the sequence is converge, please justify to the best of your ability the claimed limit in your answer. If you are using theorems in lecture and in the textbook} make that reference clear. (Eg. specify nameumber of the theorem and section of the hook.) Problem 3. For each statement, justify Whether they are true or explain why they are false (providing,r a counterexample). Each item is worth 5 points. (a) If (an) > 0 then the series 2:021 an converges. (b) If the root test is inconclusive. then the ratio test is inconclusive. (e) Let (an) be a sequence of positive terms. Suppose that on = n), where f is a continuous positive decreasins function of :3: for all :1: 2 1. If the series 2.20:1 on converges then we hg ve the equality is.\" = [I x mm. H: l (d) If 21:1 an converges then 22:1 |an| converges. (e) There are series 21:1 an for which the integral test determines convergence but the root test does not. 00 (3732 Tl=1 ' Problem 4. In this problem we study the convergence of the series 5 := E from the perspective of the different tests. Each item is worth 5 points. (i) Show that 2C 1e'" converges and its limit1 is TL: n=l . (ii) Deduce from (i) that S is convergent by comparing it to 2:021 e' . (iii) Use the integral test to show that 5' converges. Hint: You may use the beeutzful equality If; (332.19: = . (iv) Use the ratio test to show that S converges. (v) Use the root test to show that S converges