Calculus Examples

$\sum_{k = 0}^{10} 4 {(\frac{1}{2})}^{k}$

Step 1

The sum of a finite geometric series can be found using the formula $a (\frac{1 - r^{n}}{1 - r})$ where $a$ is the first term and $r$ is the ratio between successive terms.

Step 2

Find the ratio of successive terms by plugging into the formula $r = \frac{a_{k + 1}}{a_{k}}$ and simplifying.

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Step 2.1

Substitute $a_{k}$ and $a_{k + 1}$ into the formula for $r$ .

$r = \frac{4 {(\frac{1}{2})}^{k + 1}}{4 {(\frac{1}{2})}^{k}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{4 {(\frac{1}{2})}^{k + 1}}{4 {(\frac{1}{2})}^{k}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(\frac{1}{2})}^{k + 1}}{{(\frac{1}{2})}^{k}}$

Step 2.2.2

Cancel the common factor of ${(\frac{1}{2})}^{k + 1}$ and ${(\frac{1}{2})}^{k}$ .

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Step 2.2.2.1

Factor ${(\frac{1}{2})}^{k}$ out of ${(\frac{1}{2})}^{k + 1}$ .

$r = \frac{{(\frac{1}{2})}^{k} \frac{1}{2}}{{(\frac{1}{2})}^{k}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{1}{2})}^{k} \frac{1}{2}}{{(\frac{1}{2})}^{k} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{1}{2})}^{k} \frac{1}{2}}{{(\frac{1}{2})}^{k} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{\frac{1}{2}}{1}$

Step 2.2.2.2.4

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

$r = \frac{1}{2}$

Step 3

Find the first term in the series by substituting in the lower bound and simplifying.

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Step 3.1

Substitute $0$ for $k$ into $4 {(\frac{1}{2})}^{k}$ .

$a = 4 {(\frac{1}{2})}^{0}$

Step 3.2

Simplify.

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Step 3.2.1

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

$a = 4 \frac{1^{0}}{2^{0}}$

Step 3.2.2

Anything raised to $0$ is $1$ .

$a = 4 \frac{1}{2^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 4 (\frac{1}{1})$

Step 3.2.4

Cancel the common factor of $1$ .

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Step 3.2.4.1

Cancel the common factor.

$a = 4 (\frac{1}{1})$

Step 3.2.4.2

Rewrite the expression.

$a = 4 \cdot 1$

Step 3.2.5

Multiply $4$ by $1$ .

$a = 4$

Step 4

Substitute the values of the ratio, first term, and number of terms into the sum formula.

$4 \frac{1 - {(\frac{1}{2})}^{11}}{1 - \frac{1}{2}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{1}{2})}^{11}}{1 - \frac{1}{2}}$ by $\frac{2}{2}$ .

$4 (\frac{2}{2} \cdot \frac{1 - {(\frac{1}{2})}^{11}}{1 - \frac{1}{2}})$

Step 5.1.2

Combine.

$4 \frac{2 (1 - {(\frac{1}{2})}^{11})}{2 (1 - \frac{1}{2})}$

Step 5.2

Apply the distributive property.

$4 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{11})}{2 \cdot 1 + 2 (- \frac{1}{2})}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

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

$4 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{11})}{2 \cdot 1 + 2 (\frac{- 1}{2})}$

Step 5.3.2

Cancel the common factor.

$4 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{11})}{2 \cdot 1 + 2 (\frac{- 1}{2})}$

Step 5.3.3

Rewrite the expression.

$4 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{11})}{2 \cdot 1 - 1}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $2$ by $1$ .

$4 \frac{2 + 2 (- {(\frac{1}{2})}^{11})}{2 \cdot 1 - 1}$

Step 5.4.2

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

$4 \frac{2 + 2 (- \frac{1^{11}}{2^{11}})}{2 \cdot 1 - 1}$

Step 5.4.3

Cancel the common factor of $2$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{1^{11}}{2^{11}}$ into the numerator.

$4 \frac{2 + 2 \frac{- 1^{11}}{2^{11}}}{2 \cdot 1 - 1}$

Step 5.4.3.2

Factor $2$ out of $2^{11}$ .

$4 \frac{2 + 2 \frac{- 1^{11}}{2 \cdot 2^{10}}}{2 \cdot 1 - 1}$

Step 5.4.3.3

Cancel the common factor.

$4 \frac{2 + 2 \frac{- 1^{11}}{2 \cdot 2^{10}}}{2 \cdot 1 - 1}$

Step 5.4.3.4

Rewrite the expression.

$4 \frac{2 + \frac{- 1^{11}}{2^{10}}}{2 \cdot 1 - 1}$

Step 5.4.4

One to any power is one.

$4 \frac{2 + \frac{- 1 \cdot 1}{2^{10}}}{2 \cdot 1 - 1}$

Step 5.4.5

Raise $2$ to the power of $10$ .

$4 \frac{2 + \frac{- 1 \cdot 1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.6

Multiply $- 1$ by $1$ .

$4 \frac{2 + \frac{- 1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.7

Move the negative in front of the fraction.

$4 \frac{2 - \frac{1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.8

To write $2$ as a fraction with a common denominator, multiply by $\frac{1024}{1024}$ .

$4 \frac{2 \cdot \frac{1024}{1024} - \frac{1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.9

Combine $2$ and $\frac{1024}{1024}$ .

$4 \frac{\frac{2 \cdot 1024}{1024} - \frac{1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.10

Combine the numerators over the common denominator.

$4 \frac{\frac{2 \cdot 1024 - 1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $2$ by $1024$ .

$4 \frac{\frac{2048 - 1}{1024}}{2 \cdot 1 - 1}$

Step 5.4.11.2

Subtract $1$ from $2048$ .

$4 \frac{\frac{2047}{1024}}{2 \cdot 1 - 1}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $2$ by $1$ .

$4 \frac{\frac{2047}{1024}}{2 - 1}$

Step 5.5.2

Subtract $1$ from $2$ .

$4 \frac{\frac{2047}{1024}}{1}$

Step 5.6

Divide $\frac{2047}{1024}$ by $1$ .

$4 (\frac{2047}{1024})$

Step 5.7

Cancel the common factor of $4$ .

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Step 5.7.1

Factor $4$ out of $1024$ .

$4 \frac{2047}{4 (256)}$

Step 5.7.2

Cancel the common factor.

$4 \frac{2047}{4 \cdot 256}$

Step 5.7.3

Rewrite the expression.

$\frac{2047}{256}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2047}{256}$

Decimal Form:

$7.99609375$

Mixed Number Form:

$7 \frac{255}{256}$