What is expand (1-1/x)^4 ?

The solution to expand (1-1/x)^4 is 1-4/x+6/(x^2)-4/(x^3)+1/(x^4)

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expand (1-1/x)^4

expand

(1 - \frac{1}{x})^{4}

1 - \frac{4}{x} + \frac{6}{x ^{2}} - \frac{4}{x ^{3}} + \frac{1}{x ^{4}}

Solution steps

(1 - \frac{1}{x})^{4}

Apply binomial theorem:

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

a = 1, b = - \frac{1}{x}

= i = 0 \sum 4 (i 4) \cdot 1^{(4 - i)} (- \frac{1}{x})^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} (- \frac{1}{x})^{0} : 1

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (- \frac{1}{x})^{1} : - \frac{4}{x}

Simplify

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

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (- \frac{1}{x})^{3} : - \frac{4}{x ^{3}}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} (- \frac{1}{x})^{4} : \frac{1}{x ^{4}}

= 1 - \frac{4}{x} + \frac{6}{x ^{2}} - \frac{4}{x ^{3}} + \frac{1}{x ^{4}}

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\frac{( a + 2 ) ! + a ! + ( a + 1 ) !}{( a + 2 ) \cdot a !}

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What is expand (1-1/x)^4 ?
The solution to expand (1-1/x)^4 is 1-4/x+6/(x^2)-4/(x^3)+1/(x^4)