展開する ((n+1)^2sin(1/(n+1)))/(n^2sin(1/n))

解

展開する

\frac{( n + 1 ) ^{2} sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

\frac{sin ( \frac{1}{n + 1} )}{sin ( \frac{1}{n} )} + \frac{2 sin ( \frac{1}{n + 1} )}{n sin ( \frac{1}{n} )} + \frac{sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

解答ステップ

\frac{( n + 1 ) ^{2} sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

(n + 1)^{2} = n^{2} + 2 n + 1

= \frac{sin ( \frac{1}{n + 1} ) ( n ^{2} + 2 n + 1 )}{n ^{2} sin ( \frac{1}{n} )}

拡張

(n^{2} + 2 n + 1) sin (\frac{1}{n + 1}) : n^{2} sin (\frac{1}{n + 1}) + 2 n sin (\frac{1}{n + 1}) + sin (\frac{1}{n + 1})

= \frac{n ^{2} sin ( \frac{1}{n + 1} ) + 2 n sin ( \frac{1}{n + 1} ) + sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

分数の規則を適用する:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{n ^{2} sin ( \frac{1}{n + 1} ) + 2 n sin ( \frac{1}{n + 1} ) + sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} = \frac{n ^{2} sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} + \frac{2 n sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} + \frac{sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

= \frac{n ^{2} sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} + \frac{2 n sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} + \frac{sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

キャンセル

\frac{n ^{2} sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} : \frac{sin ( \frac{1}{n + 1} )}{sin ( \frac{1}{n} )}

= \frac{sin ( \frac{1}{n + 1} )}{sin ( \frac{1}{n} )} + \frac{2 n sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} + \frac{sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}

キャンセル

\frac{2 n sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} : \frac{2 sin ( \frac{1}{n + 1} )}{n sin ( \frac{1}{n} )}

= \frac{sin ( \frac{1}{n + 1} )}{sin ( \frac{1}{n} )} + \frac{2 sin ( \frac{1}{n + 1} )}{n sin ( \frac{1}{n} )} + \frac{sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )}