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

Lösung

erweitern

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

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

Schritte zur Lösung

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

(n + 1)^{2} = n^{2} + 2 n + 1

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

Multipliziere aus

(n^{2} + 2 n + 1) sin (\frac{1}{n + 1}) : n^{2} sin (\frac{1}{n + 1}) + 2 n sin (\frac{1}{n + 1}) + sin (\frac{1}{n + 1})

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

Wende Bruchregel an:

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

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

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

Streiche

\frac{n ^{2} sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} : \frac{sin ( \frac{1}{n + 1} )}{sin ( \frac{1}{n} )}

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

Streiche

\frac{2 n sin ( \frac{1}{n + 1} )}{n ^{2} sin ( \frac{1}{n} )} : \frac{2 sin ( \frac{1}{n + 1} )}{n sin ( \frac{1}{n} )}

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

e x p an d \frac{6 x + 5}{( 2 x - 1 ) ^{2}}

s im pl i f y (x - \frac{1}{2}) \cdot (x - \frac{1}{3})

e x p an d \frac{s ^{2} + ( 1 - e ^{- π s} ) s + 16}{( s ^{2} + 16 ) ^{2}}

e x p an d (2 x y - 3 x^{2}) d x + (x^{2} - 2 y) d y

e x p an d \frac{( r e ^{- x} )}{( e ^{- x} - 1 ) ^{2}}