where Hn Define yn = Hn -In(n). The limit lim (H-ln(n)) = y is known as...

 

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where Hn Define yn = Hn -In(n). The limit lim (H-ln(n)) = y is known as the Euler-Mascheroni constant 14x 72 is the n-th harmonic number. Show the sequence converges by using three steps: (a) Show that Yn Yn-1, Vn 1. (Hint: Observe Yn Yn-1 < 0 by using the Maclaurin series of ln(1-x) where set = 1/n.) n-1 k+1 (b) Given that ln(n) = [*** dt], conclude that %> 0 by showing that H-1 In(n) > 0. (Hint: k=1 Use the comparison e/k > 1 + 1.) (c) Conclude by using Monotone Bounded Sequence Theorem. where Hn Define yn = Hn -In(n). The limit lim (H-ln(n)) = y is known as the Euler-Mascheroni constant 14x 72 is the n-th harmonic number. Show the sequence converges by using three steps: (a) Show that Yn Yn-1, Vn 1. (Hint: Observe Yn Yn-1 < 0 by using the Maclaurin series of ln(1-x) where set = 1/n.) n-1 k+1 (b) Given that ln(n) = [*** dt], conclude that %> 0 by showing that H-1 In(n) > 0. (Hint: k=1 Use the comparison e/k > 1 + 1.) (c) Conclude by using Monotone Bounded Sequence Theorem.

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This question is about proving the convergence of a sequence that leads to the EulerMascheroni constant using several steps Lets go through each part ... View the full answer