erweitern (x+1/(x^2))^7

Lösung

erweitern

(x + \frac{1}{x ^{2}})^{7}

x^{7} + 7 x^{4} + 21 x + \frac{35}{x ^{2}} + \frac{35}{x ^{5}} + \frac{21}{x ^{8}} + \frac{7}{x ^{11}} + \frac{1}{x ^{14}}

Schritte zur Lösung

(x + \frac{1}{x ^{2}})^{7}

Wende Binomialsetz an:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = \frac{1}{x ^{2}}

= i = 0 \sum 7 (i 7) x^{(7 - i)} (\frac{1}{x ^{2}})^{i}

Erweitere Summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (\frac{1}{x ^{2}})^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (\frac{1}{x ^{2}})^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (\frac{1}{x ^{2}})^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (\frac{1}{x ^{2}})^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (\frac{1}{x ^{2}})^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (\frac{1}{x ^{2}})^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (\frac{1}{x ^{2}})^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (\frac{1}{x ^{2}})^{7}

Vereinfache

\frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (\frac{1}{x ^{2}})^{0} : x^{7}

Vereinfache

\frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (\frac{1}{x ^{2}})^{1} : 7 x^{4}

Vereinfache

\frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (\frac{1}{x ^{2}})^{2} : 21 x

Vereinfache

\frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (\frac{1}{x ^{2}})^{3} : \frac{35}{x ^{2}}

Vereinfache

\frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (\frac{1}{x ^{2}})^{4} : \frac{35}{x ^{5}}

Vereinfache

\frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (\frac{1}{x ^{2}})^{5} : \frac{21}{x ^{8}}

Vereinfache

\frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (\frac{1}{x ^{2}})^{6} : \frac{7}{x ^{11}}

Vereinfache

\frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (\frac{1}{x ^{2}})^{7} : \frac{1}{x ^{14}}

= x^{7} + 7 x^{4} + 21 x + \frac{35}{x ^{2}} + \frac{35}{x ^{5}} + \frac{21}{x ^{8}} + \frac{7}{x ^{11}} + \frac{1}{x ^{14}}

e x p an d (e^{3 x} + 3 e^{3 x} x) e^{3 x}

s im pl i f y (x + 1) (x - \frac{1}{2})

e x p an d \frac{s - 6}{( s ^{2} - 1 ) ( s ^{2} + 4 )}

e x p an d sin (6) (2 e^{5 x^{4}} x + 20 e^{5 x^{4}} x^{5})

e x p an d \frac{x}{( x + 3 ) ( x + 1 )}