1. If an integer n equals 2.k and k is an integer, then n is even. O equals 2.0 and O is an integer.

2. Use the Comparison Test or Limit Comparison Test to determine whether each series is convergent or divergent. Carefully explain your justification for using each test.

Determine whether each series is convergent or divergent. For each alternating series that is convergent, determine whether it is conditionally or absolutely convergent.

A 4 (a) Determine all the elements of Z35. (b) Show that U = {1, 8, 22, 29} a subgroup of Z35 is, so (U, O) < (Z85, 0). (c) Find a subgroup of Z35 that contains 11 as an element. (1+2+1 Points) Please justify all answers. Solutions without a solution path will not be evaluated.

(a) Draw and shade in the rectangles needed to represent a left sum approximation of the distance traveled on the interval [0,30] with n = 6 rectangles. (b) Use the graph to estimate the distance traveled on the interval [0,30] with a left Riemann sum with n = 6 rectangles. Note: 1 second is equal to 1/3600 hours. (c) Does this give an overestimate or an underestimate of the distance traveled? (d) Interpret the Riemann Sum in the context of the situation. Include units in your answer.

(2 + 2 Points) A 3 Let Pk: Z10 -> Z10 be an illustration with the rule Pk (n) = (k n) mod 10, k N. The figure yr is an encryption of Z10 if yr is injective. (a) Alice would like to send her friend Bob her phone number 0152-347896. For security reasons, she would like to encrypt the number and add a Find encryption of the form pr. First, she tries to do this with the- educations y2 and p3. What is your phone number in each case, after you click on every digit that has applied figures y2 and p3 respectively? (b) Show that v3 is an encryption. For which k € Z10 N N is the decoding (i.e. reverse graphing) of y3? (c) Why is 22 not an encryption? (d) For which k € N is yr an encryption and for which k € N is yk no encryption? Justify your answer. 4k (1+2+1 +2 Points) For the summer semester beginners: A figure f: X-> Y is called injective if there are no two x values that are mapped to the same y value: for all x1 # x2 € x, f(x1) # f(x2) apply. An equivalent formulation is: if f(x1) = f(x2) for two values x1, x2 € X, then x1 must be = x2 apply.

(1 + 3 Points) A 2 (a) Calculate the expression 1 @ 2@...@(n-2) @ (n-1) in the group (Zn,) for (i) n € {3,4,5,6) (ii) n N, n > 2. (b) Terence Tao's birthday is on July 17th, which is a Monday in 2023. To what days of the week is his/her birthday in 2050 and 2010? Justify the answer in a group-theoretical way. (2 + 2 Points)

(a) By means of induction according to n, show that an 2 2 holds for all n 21. (b) Show that the sequence is monotonically decreasing. (c) Show that the sequence converges and determine its limit value.

Problem 4 Consider the matrix = (ad). A = = When a + b = c+d, show that the vector (1, 1)T is an eigenvector of A and calculate both eigenvalues of A in terms of a, b, c, d.

2. For the functions f and g defined by, f(x) = cosh (3x), g(x) = log(x) find the composite functions fog and go f stating clearly the intended domain and range.

Determine whether each series is convergent or divergent. For each alternating series that is convergent, determine whether it is conditionally or absolutely convergent.