Solve 1/x+1/x^2+1/x^3+1/x^4+frac{1}{x^5}+frac{1}{{x}^6}

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

Evaluate

$$\frac{\left(x+1\right)\left(\left(x^{2}+1\right)^{2}-x^{2}\right)}{x^{6}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x^{2} is x^{2}. Multiply \frac{1}{x} times \frac{x}{x}.

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

Since \frac{x}{x^{2}} and \frac{1}{x^{2}} have the same denominator, add them by adding their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x^{3} is x^{3}. Multiply \frac{x+1}{x^{2}} times \frac{x}{x}.

$$\frac{\left(x+1\right)x}{x^{3}}+\frac{1}{x^{3}}+\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}$$

Since \frac{\left(x+1\right)x}{x^{3}} and \frac{1}{x^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)x+1}{x^{3}}+\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}$$

Do the multiplications in \left(x+1\right)x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x^{4} is x^{4}. Multiply \frac{x^{2}+x+1}{x^{3}} times \frac{x}{x}.

$$\frac{\left(x^{2}+x+1\right)x}{x^{4}}+\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}$$

Since \frac{\left(x^{2}+x+1\right)x}{x^{4}} and \frac{1}{x^{4}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+x+1\right)x+1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}$$

Do the multiplications in \left(x^{2}+x+1\right)x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{4} and x^{5} is x^{5}. Multiply \frac{x^{3}+x^{2}+x+1}{x^{4}} times \frac{x}{x}.

$$\frac{\left(x^{3}+x^{2}+x+1\right)x}{x^{5}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}$$

Since \frac{\left(x^{3}+x^{2}+x+1\right)x}{x^{5}} and \frac{1}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{3}+x^{2}+x+1\right)x+1}{x^{5}}+\frac{1}{x^{6}}$$

Do the multiplications in \left(x^{3}+x^{2}+x+1\right)x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{5} and x^{6} is x^{6}. Multiply \frac{x^{4}+x^{3}+x^{2}+x+1}{x^{5}} times \frac{x}{x}.

$$\frac{\left(x^{4}+x^{3}+x^{2}+x+1\right)x}{x^{6}}+\frac{1}{x^{6}}$$

Since \frac{\left(x^{4}+x^{3}+x^{2}+x+1\right)x}{x^{6}} and \frac{1}{x^{6}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{4}+x^{3}+x^{2}+x+1\right)x+1}{x^{6}}$$

Do the multiplications in \left(x^{4}+x^{3}+x^{2}+x+1\right)x+1.

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

Factor

$$\frac{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{2}+x+1\right)}{xx^{5}}$$