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This is an individual exam. You are not allowed to consult any person nor material. You are only allowed to a bring a writing tool. Please use the provided blue books for your writing. There are 6 i... ndependent problems totaling 110 points. A full score is considered 100 points. You have 80 minutes. Problem 1 (10 points) Find a formula for the following sum X N n=0 c n , c ∈ (0,∞) \ {1}, N ∈ {0, 1, 2, ...}, and use induction to prove the correctness of your formula. Problem 2 (10 points) Find all values of x ∈ (0,∞) — if any — for which the following limit exists in (0,∞): limn→∞ x n ? For those values of x ∈ (0,∞) with a limit in (0,∞), report the limit. Problem 3 (30 points) Consider the functions f : N −→ Q, f(n) := n 2 n2 + 1 , g : Q −→ Q, g(z) := z 2 + 2. 1. What are the domain and codomain for the composition g(f(n))? ♦ 2. Is the function f injective (1-1)? ♦ 3. Is the function g surjective (onto)? ♦ Problem 4 (10 points) Determine conditions on m ∈ Z such that the map f : Z → Z defined by f(x) := mx is bijective and prove your answer. 1 Problem 5 (30 points) For the following subquestions (a)-(f) no argumentation is needed; just clearly answer YES, NO, or write BLANK. Each correct answer gives 5 points, each incorrect answer gives -5 points, and each blank answer gives 0 points. It is not recommended to guess [Show More]

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