展開する (ln(1+1/x)-1/(x+1))(1+1/x)^x

解

展開する

(ln (1 + \frac{1}{x}) - \frac{1}{x + 1}) (1 + \frac{1}{x})^{x}

ln (1 + \frac{1}{x}) (1 + \frac{1}{x})^{x} - \frac{( x + 1 ) ^{x - 1}}{x ^{x}}

解答ステップ

(ln (1 + \frac{1}{x}) - \frac{1}{x + 1}) (1 + \frac{1}{x})^{x}

分配法則を適用する:

a (b - c) = ab - a c

a = (1 + \frac{1}{x})^{x}, b = ln (1 + \frac{1}{x}), c = \frac{1}{x + 1}

= (1 + \frac{1}{x})^{x} ln (1 + \frac{1}{x}) - (1 + \frac{1}{x})^{x} \frac{1}{x + 1}

= ln (1 + \frac{1}{x}) (1 + \frac{1}{x})^{x} - \frac{1}{x + 1} (1 + \frac{1}{x})^{x}

\frac{1}{x + 1} (1 + \frac{1}{x})^{x} = \frac{( x + 1 ) ^{x - 1}}{x ^{x}}

= ln (1 + \frac{1}{x}) (1 + \frac{1}{x})^{x} - \frac{( x + 1 ) ^{x - 1}}{x ^{x}}