What is expand 1/((n-1)^2)+1/(n-1)-1/n ?

The solution to expand 1/((n-1)^2)+1/(n-1)-1/n is (2n-1)/(n^3-2n^2+n)

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expand 1/((n-1)^2)+1/(n-1)-1/n

expand

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

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

Solution steps

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

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

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

Factor

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

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

Least Common Multiplier of

(n - 1)^{2}, n - 1, n : n (n - 1)^{2}

Adjust Fractions based on the LCM

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

Since the denominators are equal, combine the fractions:

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

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

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

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

Expand

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

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

Expand

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

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

(9 + cos (θ))^{3}

\frac{( x + 2 )}{x ( x + 4 )}

\frac{10 ( 29 e ^{30} + 1 )}{3 e ^{30}}

\frac{( 3 x + 1 ) ^{3}}{( 2 x - 1 ) ^{2} x}

2 \cdot (\frac{5}{2} x + \frac{1}{2})

What is expand 1/((n-1)^2)+1/(n-1)-1/n ?
The solution to expand 1/((n-1)^2)+1/(n-1)-1/n is (2n-1)/(n^3-2n^2+n)