Difference between revisions of "2018 Putnam B Problems/Problem 2"
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Let \( n \) be a positive integer, and let \( f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1} \). Prove that \( f_n \) has no roots in the closed unit disk \( \{z \in \mathbb{C}: |z| \leq 1 \} \). | Let \( n \) be a positive integer, and let \( f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1} \). Prove that \( f_n \) has no roots in the closed unit disk \( \{z \in \mathbb{C}: |z| \leq 1 \} \). | ||
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| + | ==Solution== | ||
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| + | {{MAA Notice}} | ||
Revision as of 16:56, 18 August 2025
Problem
Let \( n \) be a positive integer, and let \( f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1} \). Prove that \( f_n \) has no roots in the closed unit disk \( \{z \in \mathbb{C}: |z| \leq 1 \} \).
Solution
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
