Calculus Examples

$\sum_{n = 0}^{3} {(- \frac{1}{2})}^{n} \cdot (4)$

Step 1

Expand the series for each value of $n$ .

${(- \frac{1}{2})}^{0} \cdot 4 + {(- \frac{1}{2})}^{1} \cdot 4 + {(- \frac{1}{2})}^{2} \cdot 4 + {(- \frac{1}{2})}^{3} \cdot 4$

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 $- \frac{1}{2}$ .

${(- 1)}^{0} {(\frac{1}{2})}^{0} \cdot 4 + {(- \frac{1}{2})}^{1} \cdot 4 + {(- \frac{1}{2})}^{2} \cdot 4 + {(- \frac{1}{2})}^{3} \cdot 4$

Step 2.1.1.2

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

${(- 1)}^{0} \frac{1^{0}}{2^{0}} \cdot 4 + {(- \frac{1}{2})}^{1} \cdot 4 + {(- \frac{1}{2})}^{2} \cdot 4 + {(- \frac{1}{2})}^{3} \cdot 4$

Step 2.1.2

Anything raised to $0$ is $1$ .

$1 \frac{1^{0}}{2^{0}} \cdot 4 + {(- \frac{1}{2})}^{1} \cdot 4 + {(- \frac{1}{2})}^{2} \cdot 4 + {(- \frac{1}{2})}^{3} \cdot 4$

Step 2.1.3

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

$\frac{1^{0}}{2^{0}} \cdot 4 + {(- \frac{1}{2})}^{1} \cdot 4 + {(- \frac{1}{2})}^{2} \cdot 4 + {(- \frac{1}{2})}^{3} \cdot 4$

Step 2.1.4

Anything raised to $0$ is $1$ .

$\frac{1}{2^{0}} \cdot 4 + {(- \frac{1}{2})}^{1} \cdot 4 + {(- \frac{1}{2})}^{2} \cdot 4 + {(- \frac{1}{2})}^{3} \cdot 4$

Step 2.1.5

Anything raised to $0$ is $1$ .

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

Step 2.1.6

Cancel the common factor of $1$ .

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Step 2.1.6.1

Cancel the common factor.

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

Step 2.1.6.2

Rewrite the expression.

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

Step 2.1.7

Multiply $4$ by $1$ .

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

Step 2.1.8

Simplify.

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

Step 2.1.9

Cancel the common factor of $2$ .

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Step 2.1.9.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.1.9.2

Factor $2$ out of $4$ .

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

Step 2.1.9.3

Cancel the common factor.

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

Step 2.1.9.4

Rewrite the expression.

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

Step 2.1.10

Multiply $- 1$ by $2$ .

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

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 $- \frac{1}{2}$ .

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

Step 2.1.11.2

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

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

Step 2.1.12

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

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

Step 2.1.13

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

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

Step 2.1.14

One to any power is one.

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

Step 2.1.15

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

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

Step 2.1.16

Cancel the common factor of $4$ .

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Step 2.1.16.1

Cancel the common factor.

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

Step 2.1.16.2

Rewrite the expression.

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

Step 2.1.17

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

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Step 2.1.17.1

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

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

Step 2.1.17.2

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

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

Step 2.1.18

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

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

Step 2.1.19

One to any power is one.

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

Step 2.1.20

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

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

Step 2.1.21

Cancel the common factor of $4$ .

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Step 2.1.21.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

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

Step 2.1.21.2

Factor $4$ out of $8$ .

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

Step 2.1.21.3

Cancel the common factor.

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

Step 2.1.21.4

Rewrite the expression.

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

Step 2.1.22

Move the negative in front of the fraction.

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

Step 2.2

Find the common denominator.

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Step 2.2.1

Write $4$ as a fraction with denominator $1$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Write $- 2$ as a fraction with denominator $1$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $4$ by $2$ .

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

Step 2.4.2

Multiply $- 2$ by $2$ .

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

Step 2.5

Subtract $4$ from $8$ .

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

Step 2.6

Add $4$ and $2$ .

$\frac{6 - 1}{2}$

Step 2.7

Subtract $1$ from $6$ .

$\frac{5}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$