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**Q1:**Determine whether each series is convergent or divergent. For each alternating series that is convergent, determine whether it is conditionally or absolutely convergent.See Answer**Q2:**(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)See Answer**Q3:**(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.See Answer**Q4:**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.See Answer**Q5:**1. Prove each of the following for a metric space (M, d): (i) The following two statements are equivalent: (a) x € M is not isolated; (b) Every neighborhood of x contains an infinite number of points of M. (ii) If M has the property that every intersection of open sets is open, then M is discrete. (iii) If M is an infinite metric space, then M contains an infinite open set U such that both U and its complement are infinite.See Answer**Q6:**2. Prove that a metric space X is discrete if and only if every function on X to an arbitrary second metric space is continuous.See Answer**Q7:**3. Let X be a dense subset of a metric space Y. Suppose that every Cauchy sequence in X converges to a point in Y. Prove that Y is complete.See Answer**Q8:**1. Explain why the Comparison Test can or cannot be used to decide if the series converges or diverges.See Answer**Q9:**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.See Answer**Q10:**3. Explain why the Integral Test can or cannot be used to determine whether the series is convergent.See Answer**Q11:**4. The Reimann zeta-function is defined by and is used in number theory to study the distribution of primes. (a) For values of a does converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4See Answer**Q12:**5. Recall from class this week we studied the: (a) State the hypotheses (or conditions) for the Integral Test. (b) State the conclusion for the Integral Test. (c) Give a brief explanation for why each hypothesis (or condition) is necessary for the Integral Test. (d) Explain what the phrase "if and only if" means in the Integral Test.See Answer**Q13:**6 Given the following arguments for the convergence or divergence of the series. Determine if each argument is valid or invalid. Explain your reasoning.See Answer**Q14:**Show that if o<a<b, then 1-a/b< log b/a<b/a-1 deduce that 1/6 < log 1.2 <1/5See Answer**Q15:**Let f be a bounded function. The oscillation (osc) of f in ECD(f) domain of f where ESee Answer**Q16:**Let z= {x ε r: f(x)=0} f is a continuous function. Show thatSee Answer**Q17:**(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.See Answer**Q18:**A 2 Show (e.g. by induction, but there is another way) that for q € R, q # 1 See Answer**Q19:**A 3 Determine for four of the following series whether they are convergent, absolutely convergent or are divergent:See Answer**Q20:**A 4 (a) Let (an)nen and (ba)nen be two sequences in R such that the series [n=1, and [mb2 are absolutely convergent. Show that then also the seriesSee Answer

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