3_Solutions

# 3_Solutions - Assignment 3 Solutions 1 Use induction to...

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Assignment 3 – Solutions 1. Use induction to prove that 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + ··· + 1 2 n 1 + n 2 . [This can be used to prove that the inﬁnite harmonic series X r =1 1 r diverges.] Solution: For n = 1, 1 + 1 2 = 1 + 1 2 so the equality holds. As the induction hypothesis, assume that the result is true for n = k ; that is 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + ··· + 1 2 k 1 + k 2 . Then, for n = k + 1, 1 1 + 1 2 + ··· + 1 2 k + 1 2 k + 1 + ··· + 1 2 k +1 ± 1 + k 2 ² + 1 2 k + 1 + 1 2 k + 2 + ··· + 1 2 k +1 ± 1 + k 2 ² + 1 2 k +1 + 1 2 k +1 + ··· + 1 2 k +1 ± 1 + k 2 ² + 2 k 2 k +1 = 1 + k 2 + 1 2 = 1 + k + 1 2 . Hence the result holds for n = k + 1 whenever it is true for n = k . By the Principle of Mathematical Induction the result holds for all n P . 2. Let a 1 = 2 and a n +1 = 5 a n - 4 a n for n 1. Prove that for all n 1, 1 a n a n +1 4 . Solution:

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3_Solutions - Assignment 3 Solutions 1 Use induction to...

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