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

Lösung

erweitern

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

Wende Bruchregel an:

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

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

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

Streiche

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

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

Streiche

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

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

Streiche

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

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

e x p an d \frac{3 x ( x + 2 )}{( x - 4 ) ( x + 4 )}

e x p an d \frac{1}{( 2 - b ) ( - 2 + b )}

\frac{\partial}{\partial x} ((y) (x^{2} + 2 y^{2}))

e x p an d \frac{( 1 + z ^{2} ) 3}{2}

\frac{\partial}{\partial x} ((z) (x^{2} - 3 x y + y^{2}))