h3 - [30] Homework 3. Programming Assignment The goal of...

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Unformatted text preview: [30] Homework 3. Programming Assignment The goal of this assignment is to find good approximations of n (2) Hn = 1 k2 k=1 and n! = 1 · 2 · 3 · · · n for large n (n → ∞). (2) [15] Evaluation of (Zeta Function) Hn : (2) Tabulate and plot Hn for a range of n (e.g., 1 ≤ n ≤ 1000) using your favorite programming language (Maple or Mathematica are fine). Based on this numerical computation (2) find a good approximation of Hn for large n up to a constant term. That is, find a simple (2) function f (n) (e.g., f (n) = log n) such that Hn ≈ f (n) + constant (and this approximation you should check on your plots and/or tables). For example, your computations may √ (2) indicate that Hn ≈ n n + 3.2n + 4 (this is not the correct answer!). [15] Stirling’s Approximation: Tabulate and plot n log k k=1 for a range of n (e.g., 1 ≤ n ≤ 1000). Based on this numerical computation find a good approximation of n log n! = log k k=1 for large n up to a log n term, if possible. That is, your answer may look like this log n! ≈? n2 log n + n + 3 log n. Include your program in the homework write-up. 1 ...
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