無限の世界への一歩 - 数列の極限、無限級数無現等比級数四角い渦

学習環境

新装版数学読本4 (松坂和夫(著)、岩波書店)の第14章(無限の世界への一歩 - 数列の極限、無限級数)、14.3(無限級数)、無現等比級数の問26の解答を求めてみる。

\begin{array}{l} P_{1} = (a, 0) = a (1, 0) \\ P_{2} = (a, a r) = a (1, r) \\ P_{3} = (a - a r^{2}, a r) = a (1 - r^{2}, r) \\ P_{4} = (a - a r^{2}, a - a r^{3}) = a (1 - r^{2}, r - r^{3}) \\ P_{5} = (a - a r^{2} + a r^{4}, a - a r^{3}) = a (1 - r^{2} + r^{4}, r - r^{3}) \\ P_{6} = (a - a r^{2} + a r^{4}, a - a r^{3} + a r^{5}) = a (1 - r^{2} + r^{4}, r - r^{3} + r^{5}) \\ P_{7} = (a - a r^{2} + a r^{4} - a r^{6}, a - a r^{3} + a r^{5}) = a (1 - r^{2} + r^{4} - r^{6}, r - r^{3} + r^{5}) \end{array}

\begin{array}{l} P_{2 n} = a (\sum_{k = 1}^{n} {(- 1)}^{k - 1} r^{2 (k - 1)}, \sum_{k = 1}^{n} {(- 1)}^{k - 1} r^{2 k - 1}) \\ = a (\sum_{k = 1}^{n} {(- r^{2})}^{k - 1}, r \sum_{k = 1}^{n} {(- r^{2})}^{k - 1}) \end{array}

\lim_{n \to \infty} P_{2 n} = (\frac{a}{1 - (- r^{2})}, \frac{a r}{1 - (- r^{2})}) = (\frac{a}{1 + r^{2}}, \frac{a r}{1 + r^{2}})

\begin{array}{l} P_{2 n - 1} = a (\sum_{k = 1}^{n} {(- r^{2})}^{k - 1}, \sum_{k = 1}^{n} r {(- r^{2})}^{k - 2}) \\ \lim_{n \to \infty} P_{2 n - 1} = (\frac{a}{1 - (- r^{2})}, \frac{a r}{1 - (- r^{2})}) \\ = (\frac{a}{1 + r^{2}}, \frac{a r}{1 + r^{2}}) \end{array}

よって、点

(\frac{a}{1 + r^{2}}, \frac{a r}{1 + r^{2}})

に近づく。