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MATHEMATICS: ASSIGNMENT 8 There are two parts to this assignment: Part A and Part B. Part A is to be completed online, and Part B is to be handed in, both before 10:00 a.m. on Friday, January 29. Part A This part of the assignment will be found online, labelled Assignment8A, at webwork.elearning.ubc.ca. Part B 1. Consider the sequence an = n X 1 i=1 i ! − log n. Prove that {an } converges. (Hint: show that {an } is bounded and monotonic.) The limit, denoted γ, is called the Euler-Mascheroni constant. 2. (a) Prove that the alternating harmonic series X (−1)n−1 converges. n n≥1 2. (b) Show (by induction, or otherwise) that 2n X i=1 2n n X1 X1 (−1)i−1 = − . i i i=1 i i=1 3. Conclude, using questions 1 and 2, that the alternating harmonic series converges to log 2. 4. Determine whether XX m≥1 n≥1 Z 5. Let F (x) = x sin 0 1 converges. (m + n)2 1 1 dt. Find F 0 (0). (Hint: integrate t cos using integration by parts.) t t