Please solve these questions. Thank you

1. (10 marks total) Determine whether the following series converge conditionally, con- verge absolutely, or diverge (provide full justification for your answer): (a) (-1)". n'[v2 + (cos(n?))" jm In(n) ( b) I 3" n=2 n=1 2. (5 marks) Determine the interval of convergence of the series ) (1 - 2)" (show all n=1 Vn your steps). 3. (5 marks) Consider the sequence defined by recurrence relation do = 01 = 1 n(n - 1)a. = (n - 1)(n - 2)an-1 - (n - 3)an-2 for n 2 2. Find a formula for a,, the general term of the sequence, by computing the first few terms of the sequence until you can recognize a pattern. Then, find > a,. You can 1=0 use a fact (without proving it) that we will soon learn: e" =) n! for each r E R. 1=0 4. (12 marks total) Let / = [0, 1] and consider the following construction: remove the open interval (1/3, 2/3) (the open middle third portion of /) from I. This leaves us with the union of the two intervals [0, 1/3] and [2/3, 1]. Then, we remove the open mid- dle third from each of these intervals, resulting in the four intervals [0, 1/9], [2/9, 1/3], [2/3, 7/9], and [8/9, 1]. We repeat this process indefinitely, removing the open middle third from every interval that remains from the preceding step. The set of all numbers in [0, 1] that remain after all these open middle third intervals have been removed is called the Cantor ternary set, or simply Cantor set, named after the German math- ematician George Cantor (1845-1918). A graphical representation of the construction of the Cantor set is shown below.ii ii (a) (5 marks) Given an interval (a, b) C R with a