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This problem has an exact solution, written in the book

N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^n}{\rho^{n-2}(1-\rho^2)}\sim 2^{-n}\rho^{n+2}/(1-\rho^2),\quad n\to\infty.$$ The LHS is the exact expression of the error. The polynomial of best approximation is unique and is also described.

Remark. Akhiezer's book is available on my web page: https://www.math.purdue.edu/~eremenko/books-papers.html (Russian original). The result is on p. 70-71, and the polynomial of the best approximation is written there.

This problem has an exact solution, written in the book

N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^n}{\rho^{n-2}(1-\rho^2)}\sim 2^{-n}\rho^{n+2}/(1-\rho^2),\quad n\to\infty.$$ The LHS is the exact expression of the error. The polynomial of best approximation is unique and is also described, though the formula is too complicated to be reproduced here.

Remark. Russian original of Akhiezer's book is available on my web page: https://www.math.purdue.edu/~eremenko/books-papers.html (An English translation exists.). The result is on p. 70-71, and the polynomial of the best approximation is written there.

This problem has an exact solution, written in the book

N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^n}{\rho^{n-2}(1-\rho^2)}\sim 2^{-n}\rho^{n+2}/(1-\rho^2),\quad n\to\infty.$$ The LHS is the exact expression of the error. The polynomial of best approximation is unique and is also described.

This problem has an exact solution, written in the book

N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^n}{\rho^{n-2}(1-\rho^2)}\sim 2^{-n}\rho^{n+2}/(1-\rho^2),\quad n\to\infty.$$ The LHS is the exact expression of the error. The polynomial of best approximation is unique and is also described.

Remark. Akhiezer's book is available on my web page: https://www.math.purdue.edu/~eremenko/books-papers.html (Russian original). The result is on p. 70-71, and the polynomial of the best approximation is written there.

This problem has an exact solution, written in the book

N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^n}{\rho^{n-2}(1-\rho^2)}\sim 2^{-n+2}\rho^{n}/(1-\rho^2),\quad n\to\infty.$$ The LHS is the exact expression of the error. The polynomial of best approximation is unique and is also described.

This problem has an exact solution, written in the book

N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^n}{\rho^{n-2}(1-\rho^2)}\sim 2^{-n}\rho^{n+2}/(1-\rho^2),\quad n\to\infty.$$ The LHS is the exact expression of the error. The polynomial of best approximation is unique and is also described.