What is expand (1+1/x)^6 ?

The solution to expand (1+1/x)^6 is 1+6/x+(15)/(x^2)+(20)/(x^3)+(15)/(x^4)+6/(x^5)+1/(x^6)

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expand (1+1/x)^6

expand

(1 + \frac{1}{x})^{6}

1 + \frac{6}{x} + \frac{15}{x ^{2}} + \frac{20}{x ^{3}} + \frac{15}{x ^{4}} + \frac{6}{x ^{5}} + \frac{1}{x ^{6}}

Solution steps

(1 + \frac{1}{x})^{6}

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 6 (i 6) \cdot 1^{(6 - i)} (\frac{1}{x})^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (\frac{1}{x})^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (\frac{1}{x})^{1} : \frac{6}{x}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (\frac{1}{x})^{2} : \frac{15}{x ^{2}}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (\frac{1}{x})^{3} : \frac{20}{x ^{3}}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (\frac{1}{x})^{4} : \frac{15}{x ^{4}}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (\frac{1}{x})^{5} : \frac{6}{x ^{5}}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} (\frac{1}{x})^{6} : \frac{1}{x ^{6}}

= 1 + \frac{6}{x} + \frac{15}{x ^{2}} + \frac{20}{x ^{3}} + \frac{15}{x ^{4}} + \frac{6}{x ^{5}} + \frac{1}{x ^{6}}

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

(x^{3} e^{3 x} + 2) (x)

(12 x + 2)^{2}

(e^{9 i t} - 3 e^{3 i t} + 3 e^{- 3 i t} - e^{- 9 i t}) (e^{4 i t} + e^{- 4 i t})

(4 x^{2} + 1) 4 x^{2} + 1

What is expand (1+1/x)^6 ?
The solution to expand (1+1/x)^6 is 1+6/x+(15)/(x^2)+(20)/(x^3)+(15)/(x^4)+6/(x^5)+1/(x^6)