Calculus Examples

$\sum_{k = 0}^{3} {(- \frac{1}{2})}^{k}$

Step 1

Expand the series for each value of $k$ .

${(- \frac{1}{2})}^{0} + {(- \frac{1}{2})}^{1} + {(- \frac{1}{2})}^{2} + {(- \frac{1}{2})}^{3}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

${(- 1)}^{0} {(\frac{1}{2})}^{0} + {(- \frac{1}{2})}^{1} + {(- \frac{1}{2})}^{2} + {(- \frac{1}{2})}^{3}$

Step 2.1.1.2

Apply the product rule to $\frac{1}{2}$ .

${(- 1)}^{0} \frac{1^{0}}{2^{0}} + {(- \frac{1}{2})}^{1} + {(- \frac{1}{2})}^{2} + {(- \frac{1}{2})}^{3}$

Step 2.1.2

Anything raised to $0$ is $1$ .

$1 \frac{1^{0}}{2^{0}} + {(- \frac{1}{2})}^{1} + {(- \frac{1}{2})}^{2} + {(- \frac{1}{2})}^{3}$

Step 2.1.3

Multiply $\frac{1^{0}}{2^{0}}$ by $1$ .

$\frac{1^{0}}{2^{0}} + {(- \frac{1}{2})}^{1} + {(- \frac{1}{2})}^{2} + {(- \frac{1}{2})}^{3}$

Step 2.1.4

Anything raised to $0$ is $1$ .

$\frac{1}{2^{0}} + {(- \frac{1}{2})}^{1} + {(- \frac{1}{2})}^{2} + {(- \frac{1}{2})}^{3}$

Step 2.1.5

Anything raised to $0$ is $1$ .

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

Step 2.1.6

Divide $1$ by $1$ .

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

Step 2.1.7

Simplify.

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

Step 2.1.8

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

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Step 2.1.8.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 2.1.8.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.1.9

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

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

Step 2.1.10

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

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

Step 2.1.11

One to any power is one.

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

Step 2.1.12

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

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

Step 2.1.13

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

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Step 2.1.13.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 2.1.13.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.1.14

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

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

Step 2.1.15

One to any power is one.

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

Step 2.1.16

Raise $2$ to the power of $3$ .

$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}$

Step 2.2

Find the common denominator.

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Step 2.2.1

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

$\frac{1}{1} - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}$

Step 2.2.2

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$\frac{1}{1} \cdot \frac{8}{8} - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}$

Step 2.2.3

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$\frac{8}{8} - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}$

Step 2.2.4

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$\frac{8}{8} - (\frac{1}{2} \cdot \frac{4}{4}) + \frac{1}{4} - \frac{1}{8}$

Step 2.2.5

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$\frac{8}{8} - \frac{4}{2 \cdot 4} + \frac{1}{4} - \frac{1}{8}$

Step 2.2.6

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{8}{8} - \frac{4}{2 \cdot 4} + \frac{1}{4} \cdot \frac{2}{2} - \frac{1}{8}$

Step 2.2.7

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{8}{8} - \frac{4}{2 \cdot 4} + \frac{2}{4 \cdot 2} - \frac{1}{8}$

Step 2.2.8

Multiply $2$ by $4$ .

$\frac{8}{8} - \frac{4}{8} + \frac{2}{4 \cdot 2} - \frac{1}{8}$

Step 2.2.9

Reorder the factors of $4 \cdot 2$ .

$\frac{8}{8} - \frac{4}{8} + \frac{2}{2 \cdot 4} - \frac{1}{8}$

Step 2.2.10

Multiply $2$ by $4$ .

$\frac{8}{8} - \frac{4}{8} + \frac{2}{8} - \frac{1}{8}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{8 - 4 + 2 - 1}{8}$

Step 2.4

Subtract $4$ from $8$ .

$\frac{4 + 2 - 1}{8}$

Step 2.5

Add $4$ and $2$ .

$\frac{6 - 1}{8}$

Step 2.6

Subtract $1$ from $6$ .

$\frac{5}{8}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{8}$

Decimal Form:

$0.625$