Calculus Examples

$\sum_{0}^{8} 8 {(- \frac{2}{3})}^{i}$

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 $r = \frac{a_{i + 1}}{a_{i}}$ and simplifying.

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

$r = \frac{8 {(- \frac{2}{3})}^{i + 1}}{8 {(- \frac{2}{3})}^{i}}$

Simplify.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$r = \frac{8 {(- \frac{2}{3})}^{i + 1}}{8 {(- \frac{2}{3})}^{i}}$

Rewrite the expression.

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

Cancel the common factor of ${(- \frac{2}{3})}^{i + 1}$ and ${(- \frac{2}{3})}^{i}$ .

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Factor ${(- \frac{2}{3})}^{i}$ out of ${(- \frac{2}{3})}^{i + 1}$ .

$r = \frac{{(- \frac{2}{3})}^{i} (- \frac{2}{3})}{{(- \frac{2}{3})}^{i}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{{(- \frac{2}{3})}^{i} (- \frac{2}{3})}{{(- \frac{2}{3})}^{i} \cdot 1}$

Cancel the common factor.

$r = \frac{{(- \frac{2}{3})}^{i} (- \frac{2}{3})}{{(- \frac{2}{3})}^{i} \cdot 1}$

Rewrite the expression.

$r = \frac{- \frac{2}{3}}{1}$

Divide $- \frac{2}{3}$ by $1$ .

$r = - \frac{2}{3}$

Step 3

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

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Substitute $0$ for $i$ into $8 {(- \frac{2}{3})}^{i}$ .

$a = 8 {(- \frac{2}{3})}^{0}$

Simplify.

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

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Apply the product rule to $- \frac{2}{3}$ .

$a = 8 ({(- 1)}^{0} {(\frac{2}{3})}^{0})$

Apply the product rule to $\frac{2}{3}$ .

$a = 8 ({(- 1)}^{0} \frac{2^{0}}{3^{0}})$

Anything raised to $0$ is $1$ .

$a = 8 (1 \frac{2^{0}}{3^{0}})$

Multiply $\frac{2^{0}}{3^{0}}$ by $1$ .

$a = 8 \frac{2^{0}}{3^{0}}$

Anything raised to $0$ is $1$ .

$a = 8 \frac{1}{3^{0}}$

Anything raised to $0$ is $1$ .

$a = 8 (\frac{1}{1})$

Cancel the common factor of $1$ .

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Cancel the common factor.

$a = 8 (\frac{1}{1})$

Rewrite the expression.

$a = 8 \cdot 1$

Multiply $8$ by $1$ .

$a = 8$

Step 4

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

$8 \frac{1 - {(- \frac{2}{3})}^{9}}{1 - - \frac{2}{3}}$

Step 5

Simplify.

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Simplify the numerator.

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

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Apply the product rule to $- \frac{2}{3}$ .

$8 \frac{1 - ({(- 1)}^{9} {(\frac{2}{3})}^{9})}{1 - - \frac{2}{3}}$

Apply the product rule to $\frac{2}{3}$ .

$8 \frac{1 - ({(- 1)}^{9} \frac{2^{9}}{3^{9}})}{1 - - \frac{2}{3}}$

Multiply $- 1$ by ${(- 1)}^{9}$ by adding the exponents.

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Move ${(- 1)}^{9}$ .

$8 \frac{1 + {(- 1)}^{9} \cdot - 1 \frac{2^{9}}{3^{9}}}{1 - - \frac{2}{3}}$

Multiply ${(- 1)}^{9}$ by $- 1$ .

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

$8 \frac{1 + {(- 1)}^{9} \cdot {(- 1)}^{1} \frac{2^{9}}{3^{9}}}{1 - - \frac{2}{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 \frac{1 + {(- 1)}^{9 + 1} \frac{2^{9}}{3^{9}}}{1 - - \frac{2}{3}}$

Add $9$ and $1$ .

$8 \frac{1 + {(- 1)}^{10} \frac{2^{9}}{3^{9}}}{1 - - \frac{2}{3}}$

Raise $- 1$ to the power of $10$ .

$8 \frac{1 + 1 \frac{2^{9}}{3^{9}}}{1 - - \frac{2}{3}}$

Multiply $\frac{2^{9}}{3^{9}}$ by $1$ .

$8 \frac{1 + \frac{2^{9}}{3^{9}}}{1 - - \frac{2}{3}}$

Raise $2$ to the power of $9$ .

$8 \frac{1 + \frac{512}{3^{9}}}{1 - - \frac{2}{3}}$

Raise $3$ to the power of $9$ .

$8 \frac{1 + \frac{512}{19683}}{1 - - \frac{2}{3}}$

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

$8 \frac{\frac{19683}{19683} + \frac{512}{19683}}{1 - - \frac{2}{3}}$

Combine the numerators over the common denominator.

$8 \frac{\frac{19683 + 512}{19683}}{1 - - \frac{2}{3}}$

Add $19683$ and $512$ .

$8 \frac{\frac{20195}{19683}}{1 - - \frac{2}{3}}$

Simplify the denominator.

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Multiply $- - \frac{2}{3}$ .

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Multiply $- 1$ by $- 1$ .

$8 \frac{\frac{20195}{19683}}{1 + 1 (\frac{2}{3})}$

Multiply $\frac{2}{3}$ by $1$ .

$8 \frac{\frac{20195}{19683}}{1 + \frac{2}{3}}$

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

$8 \frac{\frac{20195}{19683}}{\frac{3}{3} + \frac{2}{3}}$

Combine the numerators over the common denominator.

$8 \frac{\frac{20195}{19683}}{\frac{3 + 2}{3}}$

Add $3$ and $2$ .

$8 \frac{\frac{20195}{19683}}{\frac{5}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$8 (\frac{20195}{19683} \cdot \frac{3}{5})$

Cancel the common factor of $5$ .

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Factor $5$ out of $20195$ .

$8 (\frac{5 (4039)}{19683} \cdot \frac{3}{5})$

Cancel the common factor.

$8 (\frac{5 \cdot 4039}{19683} \cdot \frac{3}{5})$

Rewrite the expression.

$8 (\frac{4039}{19683} \cdot 3)$

Cancel the common factor of $3$ .

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Factor $3$ out of $19683$ .

$8 (\frac{4039}{3 (6561)} \cdot 3)$

Cancel the common factor.

$8 (\frac{4039}{3 \cdot 6561} \cdot 3)$

Rewrite the expression.

$8 (\frac{4039}{6561})$

Multiply $8 (\frac{4039}{6561})$ .

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Combine $8$ and $\frac{4039}{6561}$ .

$\frac{8 \cdot 4039}{6561}$

Multiply $8$ by $4039$ .

$\frac{32312}{6561}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{32312}{6561}$

Decimal Form:

$4.92485901 \dots$

Mixed Number Form:

$4 \frac{6068}{6561}$