erweitern ((n+1)(n+2)(n+3))/(n^3)

Lösung

erweitern

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

1 + \frac{6}{n} + \frac{11}{n ^{2}} + \frac{6}{n ^{3}}

Schritte zur Lösung

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

Multipliziere aus

(n + 1) (n + 2) (n + 3) : n^{3} + 6 n^{2} + 11 n + 6

= \frac{n ^{3} + 6 n ^{2} + 11 n + 6}{n ^{3}}

Wende Bruchregel an:

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

\frac{n ^{3} + 6 n ^{2} + 11 n + 6}{n ^{3}} = \frac{n ^{3}}{n ^{3}} + \frac{6 n ^{2}}{n ^{3}} + \frac{11 n}{n ^{3}} + \frac{6}{n ^{3}}

= \frac{n ^{3}}{n ^{3}} + \frac{6 n ^{2}}{n ^{3}} + \frac{11 n}{n ^{3}} + \frac{6}{n ^{3}}

Wende Regel an

\frac{a}{a} = 1

\frac{n ^{3}}{n ^{3}} = 1

= 1 + \frac{6 n ^{2}}{n ^{3}} + \frac{11 n}{n ^{3}} + \frac{6}{n ^{3}}

Streiche

\frac{6 n ^{2}}{n ^{3}} : \frac{6}{n}

= 1 + \frac{6}{n} + \frac{11 n}{n ^{3}} + \frac{6}{n ^{3}}

Streiche

\frac{11 n}{n ^{3}} : \frac{11}{n ^{2}}

= 1 + \frac{6}{n} + \frac{11}{n ^{2}} + \frac{6}{n ^{3}}

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