【微分積分】3-6-2 漸近展開｜問題集
1.次の関数を\(o(x^n)\)を使って\(2\)次の項まで表示しなさい。
(1)\(3x+x^2+x^3\)
\(=3x+x^2+o(x^2)\ \ \ (x\to0)\)
(2)\(-x+3x^2+5x^3\)
\(=-x+3x^2+o(x^2)\ \ \ (x\to0)\)
(3)\(2x-4x^2+x^7\)
\(=2x-4x^2+o(x^2)\ \ \ (x\to0)\)
(4)\((1+x+3x^3+o(x^3))-(2x-6x^2+o(x^2))\)
\(=1-x+6x^2+(3x^3+o(x^3)-o(x^2))\)
\(=1-x+6x^2+o(x^2)\ \ \ (x\to0)\)
\(=1-x+6x^2+o(x^2)\ \ \ (x\to0)\)
(5)\((1+x+x^2+o(x^2))(2x-x^2+o(x^2))\)
\(=2x+(-1+2)x^2+o(x^2)\)
\(=2x+x^2+o(x^2)\ \ \ (x\to0)\)
\(=2x+x^2+o(x^2)\ \ \ (x\to0)\)
(6)\(e^x\)
\(\displaystyle =1+x+\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)
(7)\(\sin x\)
\(=x+o(x^2)\ \ \ (x\to0)\)
(8)\(\cos x\)
\(\displaystyle =1-\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)
(9)\(\log(1+x)\)
\(\displaystyle =x-\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)
次の学習に進もう！