簡約 1/((n+1)^2)+1/(n-1)-1/((n-1)^3)-1/n

解

簡約

\frac{1}{( n + 1 ) ^{2}} + \frac{1}{n - 1} - \frac{1}{( n - 1 ) ^{3}} - \frac{1}{n}

\frac{2 n ^{4} - 4 n ^{3} - n ^{2} - 2 n + 1}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}}

解答ステップ

\frac{1}{( n + 1 ) ^{2}} + \frac{1}{n - 1} - \frac{1}{( n - 1 ) ^{3}} - \frac{1}{n}

以下の最小公倍数:

(n + 1)^{2}, n - 1, (n - 1)^{3}, n : n (n + 1)^{2} (n - 1)^{3}

LCMに基づいて分数を調整する

= \frac{n ( n - 1 ) ^{3}}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}} + \frac{n ( n + 1 ) ^{2} ( n - 1 ) ^{2}}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}} - \frac{n ( n + 1 ) ^{2}}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}} - \frac{( n + 1 ) ^{2} ( n - 1 ) ^{3}}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}}

分数の規則を適用する:

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

= \frac{n ( n - 1 ) ^{3} + n ( n + 1 ) ^{2} ( n - 1 ) ^{2} - n ( n + 1 ) ^{2} - ( n + 1 ) ^{2} ( n - 1 ) ^{3}}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}}

簡素化

n (n - 1)^{3} + n (n + 1)^{2} (n - 1)^{2} - n (n + 1)^{2} - (n + 1)^{2} (n - 1)^{3} : 2 n^{4} - 4 n^{3} - n^{2} - 2 n + 1

= \frac{2 n ^{4} - 4 n ^{3} - n ^{2} - 2 n + 1}{n ( n + 1 ) ^{2} ( n - 1 ) ^{3}}