Calculus Examples

$\frac{1 - n^{- 2}}{1 - n}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - n^{- 2}}{1 - n}$

Step 1.2

Rewrite $n^{- 2}$ as ${(n^{- 1})}^{2}$ .

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = n^{- 1}$ .

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

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.2

Write $1$ as a fraction with a common denominator.

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

Step 1.4.3

Combine the numerators over the common denominator.

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

Step 1.4.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{n + 1}{n} (1 - \frac{1}{n})}{1 - n}$

Step 1.4.5

Write $1$ as a fraction with a common denominator.

$\frac{\frac{n + 1}{n} (\frac{n}{n} - \frac{1}{n})}{1 - n}$

Step 1.4.6

Combine the numerators over the common denominator.

$\frac{\frac{n + 1}{n} \cdot \frac{n - 1}{n}}{1 - n}$

Step 2

Multiply $\frac{n + 1}{n}$ by $\frac{n - 1}{n}$ .

$\frac{\frac{(n + 1) (n - 1)}{n \cdot n}}{1 - n}$

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Raise $n$ to the power of $1$ .

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

Step 3.2

Raise $n$ to the power of $1$ .

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

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{(n + 1) (n - 1)}{n^{2}}$ by $\frac{1}{1 - n}$ .

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

Step 6

Cancel the common factor of $n - 1$ and $1 - n$ .

Tap for more steps...

Step 6.1

Factor $- 1$ out of $n$ .

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

Step 6.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 6.3

Factor $- 1$ out of $- 1 (- n) - 1 (1)$ .

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

Step 6.4

Reorder terms.

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

Step 6.5

Cancel the common factor.

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

Step 6.6

Rewrite the expression.

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

Step 7

Simplify the expression.

Tap for more steps...

Step 7.1

Move $- 1$ to the left of $n + 1$ .

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

Step 7.2

Move the negative in front of the fraction.

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