erweitern ((n+1)^5)/(n^6)

Lösung

erweitern

\frac{( n + 1 ) ^{5}}{n ^{6}}

\frac{1}{n} + \frac{5}{n ^{2}} + \frac{10}{n ^{3}} + \frac{10}{n ^{4}} + \frac{5}{n ^{5}} + \frac{1}{n ^{6}}

Schritte zur Lösung

\frac{( n + 1 ) ^{5}}{n ^{6}}

(n + 1)^{5} = n^{5} + 5 n^{4} + 10 n^{3} + 10 n^{2} + 5 n + 1

= \frac{n ^{5} + 5 n ^{4} + 10 n ^{3} + 10 n ^{2} + 5 n + 1}{n ^{6}}

Wende Bruchregel an:

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

\frac{n ^{5} + 5 n ^{4} + 10 n ^{3} + 10 n ^{2} + 5 n + 1}{n ^{6}} = \frac{n ^{5}}{n ^{6}} + \frac{5 n ^{4}}{n ^{6}} + \frac{10 n ^{3}}{n ^{6}} + \frac{10 n ^{2}}{n ^{6}} + \frac{5 n}{n ^{6}} + \frac{1}{n ^{6}}

= \frac{n ^{5}}{n ^{6}} + \frac{5 n ^{4}}{n ^{6}} + \frac{10 n ^{3}}{n ^{6}} + \frac{10 n ^{2}}{n ^{6}} + \frac{5 n}{n ^{6}} + \frac{1}{n ^{6}}

\frac{n ^{5}}{n ^{6}} = \frac{1}{n}

\frac{5 n ^{4}}{n ^{6}} = \frac{5}{n ^{2}}

\frac{10 n ^{3}}{n ^{6}} = \frac{10}{n ^{3}}

\frac{10 n ^{2}}{n ^{6}} = \frac{10}{n ^{4}}

\frac{5 n}{n ^{6}} = \frac{5}{n ^{5}}

= \frac{1}{n} + \frac{5}{n ^{2}} + \frac{10}{n ^{3}} + \frac{10}{n ^{4}} + \frac{5}{n ^{5}} + \frac{1}{n ^{6}}

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