Calculus Examples

$\sum_{k = 1}^{4} {(- \frac{1}{7})}^{k}$

Step 1

Expand the series for each value of $k$ .

${(- \frac{1}{7})}^{1} + {(- \frac{1}{7})}^{2} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify.

$- \frac{1}{7} + {(- \frac{1}{7})}^{2} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.2.1

Apply the product rule to $- \frac{1}{7}$ .

$- \frac{1}{7} + {(- 1)}^{2} {(\frac{1}{7})}^{2} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.2.2

Apply the product rule to $\frac{1}{7}$ .

$- \frac{1}{7} + {(- 1)}^{2} \frac{1^{2}}{7^{2}} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.3

Raise $- 1$ to the power of $2$ .

$- \frac{1}{7} + 1 \frac{1^{2}}{7^{2}} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.4

Multiply $\frac{1^{2}}{7^{2}}$ by $1$ .

$- \frac{1}{7} + \frac{1^{2}}{7^{2}} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.5

One to any power is one.

$- \frac{1}{7} + \frac{1}{7^{2}} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.6

Raise $7$ to the power of $2$ .

$- \frac{1}{7} + \frac{1}{49} + {(- \frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.7.1

Apply the product rule to $- \frac{1}{7}$ .

$- \frac{1}{7} + \frac{1}{49} + {(- 1)}^{3} {(\frac{1}{7})}^{3} + {(- \frac{1}{7})}^{4}$

Step 2.1.7.2

Apply the product rule to $\frac{1}{7}$ .

$- \frac{1}{7} + \frac{1}{49} + {(- 1)}^{3} \frac{1^{3}}{7^{3}} + {(- \frac{1}{7})}^{4}$

Step 2.1.8

Raise $- 1$ to the power of $3$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1^{3}}{7^{3}} + {(- \frac{1}{7})}^{4}$

Step 2.1.9

One to any power is one.

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{7^{3}} + {(- \frac{1}{7})}^{4}$

Step 2.1.10

Raise $7$ to the power of $3$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + {(- \frac{1}{7})}^{4}$

Step 2.1.11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.11.1

Apply the product rule to $- \frac{1}{7}$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + {(- 1)}^{4} {(\frac{1}{7})}^{4}$

Step 2.1.11.2

Apply the product rule to $\frac{1}{7}$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + {(- 1)}^{4} \frac{1^{4}}{7^{4}}$

Step 2.1.12

Raise $- 1$ to the power of $4$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + 1 \frac{1^{4}}{7^{4}}$

Step 2.1.13

Multiply $\frac{1^{4}}{7^{4}}$ by $1$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + \frac{1^{4}}{7^{4}}$

Step 2.1.14

One to any power is one.

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + \frac{1}{7^{4}}$

Step 2.1.15

Raise $7$ to the power of $4$ .

$- \frac{1}{7} + \frac{1}{49} - \frac{1}{343} + \frac{1}{2401}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{1}{7}$ by $\frac{343}{343}$ .

$- (\frac{1}{7} \cdot \frac{343}{343}) + \frac{1}{49} - \frac{1}{343} + \frac{1}{2401}$

Step 2.2.2

Multiply $\frac{1}{7}$ by $\frac{343}{343}$ .

$- \frac{343}{7 \cdot 343} + \frac{1}{49} - \frac{1}{343} + \frac{1}{2401}$

Step 2.2.3

Multiply $\frac{1}{49}$ by $\frac{49}{49}$ .

$- \frac{343}{7 \cdot 343} + \frac{1}{49} \cdot \frac{49}{49} - \frac{1}{343} + \frac{1}{2401}$

Step 2.2.4

Multiply $\frac{1}{49}$ by $\frac{49}{49}$ .

$- \frac{343}{7 \cdot 343} + \frac{49}{49 \cdot 49} - \frac{1}{343} + \frac{1}{2401}$

Step 2.2.5

Multiply $\frac{1}{343}$ by $\frac{7}{7}$ .

$- \frac{343}{7 \cdot 343} + \frac{49}{49 \cdot 49} - (\frac{1}{343} \cdot \frac{7}{7}) + \frac{1}{2401}$

Step 2.2.6

Multiply $\frac{1}{343}$ by $\frac{7}{7}$ .

$- \frac{343}{7 \cdot 343} + \frac{49}{49 \cdot 49} - \frac{7}{343 \cdot 7} + \frac{1}{2401}$

Step 2.2.7

Multiply $7$ by $343$ .

$- \frac{343}{2401} + \frac{49}{49 \cdot 49} - \frac{7}{343 \cdot 7} + \frac{1}{2401}$

Step 2.2.8

Multiply $49$ by $49$ .

$- \frac{343}{2401} + \frac{49}{2401} - \frac{7}{343 \cdot 7} + \frac{1}{2401}$

Step 2.2.9

Reorder the factors of $343 \cdot 7$ .

$- \frac{343}{2401} + \frac{49}{2401} - \frac{7}{7 \cdot 343} + \frac{1}{2401}$

Step 2.2.10

Multiply $7$ by $343$ .

$- \frac{343}{2401} + \frac{49}{2401} - \frac{7}{2401} + \frac{1}{2401}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{- 343 + 49 - 7 + 1}{2401}$

Step 2.4

Add $- 343$ and $49$ .

$\frac{- 294 - 7 + 1}{2401}$

Step 2.5

Subtract $7$ from $- 294$ .

$\frac{- 301 + 1}{2401}$

Step 2.6

Add $- 301$ and $1$ .

$\frac{- 300}{2401}$

Step 2.7

Move the negative in front of the fraction.

$- \frac{300}{2401}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{300}{2401}$

Decimal Form:

$- 0.12494793 \dots$