Calculus Examples

$\sum_{k = 0}^{2} 12 {(- \frac{1}{2})}^{k}$

Step 1

Expand the series for each value of $k$ .

$12 {(- \frac{1}{2})}^{0} + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

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

$12 ({(- 1)}^{0} {(\frac{1}{2})}^{0}) + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.1.2

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

$12 ({(- 1)}^{0} \frac{1^{0}}{2^{0}}) + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.2

Anything raised to $0$ is $1$ .

$12 (1 \frac{1^{0}}{2^{0}}) + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.3

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

$12 \frac{1^{0}}{2^{0}} + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.4

Anything raised to $0$ is $1$ .

$12 \frac{1}{2^{0}} + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.5

Anything raised to $0$ is $1$ .

$12 (\frac{1}{1}) + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.6

Cancel the common factor of $1$ .

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Step 2.1.6.1

Cancel the common factor.

$12 (\frac{1}{1}) + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.6.2

Rewrite the expression.

$12 \cdot 1 + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.7

Multiply $12$ by $1$ .

$12 + 12 {(- \frac{1}{2})}^{1} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.8

Simplify.

$12 + 12 (- \frac{1}{2}) + 12 {(- \frac{1}{2})}^{2}$

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.

$12 + 12 (\frac{- 1}{2}) + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.9.2

Factor $2$ out of $12$ .

$12 + 2 (6) \frac{- 1}{2} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.9.3

Cancel the common factor.

$12 + 2 \cdot 6 \frac{- 1}{2} + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.9.4

Rewrite the expression.

$12 + 6 \cdot - 1 + 12 {(- \frac{1}{2})}^{2}$

Step 2.1.10

Multiply $6$ by $- 1$ .

$12 - 6 + 12 {(- \frac{1}{2})}^{2}$

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

$12 - 6 + 12 ({(- 1)}^{2} {(\frac{1}{2})}^{2})$

Step 2.1.11.2

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

$12 - 6 + 12 ({(- 1)}^{2} \frac{1^{2}}{2^{2}})$

Step 2.1.12

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

$12 - 6 + 12 (1 \frac{1^{2}}{2^{2}})$

Step 2.1.13

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

$12 - 6 + 12 \frac{1^{2}}{2^{2}}$

Step 2.1.14

One to any power is one.

$12 - 6 + 12 \frac{1}{2^{2}}$

Step 2.1.15

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

$12 - 6 + 12 (\frac{1}{4})$

Step 2.1.16

Cancel the common factor of $4$ .

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Step 2.1.16.1

Factor $4$ out of $12$ .

$12 - 6 + 4 (3) \frac{1}{4}$

Step 2.1.16.2

Cancel the common factor.

$12 - 6 + 4 \cdot 3 \frac{1}{4}$

Step 2.1.16.3

Rewrite the expression.

$12 - 6 + 3$

Step 2.2

Subtract $6$ from $12$ .

$6 + 3$

Step 2.3

Add $6$ and $3$ .

$9$