Calculus Examples

$\sum_{n = 1}^{8} - 4 {(\frac{1}{3})}^{n - 1}$

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

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Step 2.1

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

$r = \frac{- 4 {(\frac{1}{3})}^{n + 1 - 1}}{- 4 {(\frac{1}{3})}^{n - 1}}$

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.

$r = \frac{- 4 {(\frac{1}{3})}^{n + 1 - 1}}{- 4 {(\frac{1}{3})}^{n - 1}}$

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

$r = \frac{{(\frac{1}{3})}^{n - 1} {(\frac{1}{3})}^{n + 0 - (n - 1)}}{{(\frac{1}{3})}^{n - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{1}{3})}^{n - 1} {(\frac{1}{3})}^{n + 0 - (n - 1)}}{{(\frac{1}{3})}^{n - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{1}{3})}^{n - 1} {(\frac{1}{3})}^{n + 0 - (n - 1)}}{{(\frac{1}{3})}^{n - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(\frac{1}{3})}^{n + 0 - (n - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(\frac{1}{3})}^{n + 0 - (n - 1)}$ by $1$ .

$r = {(\frac{1}{3})}^{n + 0 - (n - 1)}$

Step 2.2.3

Add $n$ and $0$ .

$r = {(\frac{1}{3})}^{n - (n - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(\frac{1}{3})}^{n - n - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Subtract $n$ from $n$ .

$r = {(\frac{1}{3})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(\frac{1}{3})}^{1}$

Step 2.2.7

Simplify.

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

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 $1$ for $n$ into $- 4 {(\frac{1}{3})}^{n - 1}$ .

$a = - 4 {(\frac{1}{3})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = - 4 {(\frac{1}{3})}^{0}$

Step 3.2.2

Apply the product rule to $\frac{1}{3}$ .

$a = - 4 \frac{1^{0}}{3^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = - 4 \frac{1}{3^{0}}$

Step 3.2.4

Anything raised to $0$ is $1$ .

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

Step 3.2.5

Cancel the common factor of $1$ .

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Step 3.2.5.1

Cancel the common factor.

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

Step 3.2.5.2

Rewrite the expression.

$a = - 4 \cdot 1$

Step 3.2.6

Multiply $- 4$ by $1$ .

$a = - 4$

Step 4

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

$- 4 \frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}}$ by $\frac{3}{3}$ .

$- 4 (\frac{3}{3} \cdot \frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}})$

Step 5.1.2

Combine.

$- 4 \frac{3 (1 - {(\frac{1}{3})}^{8})}{3 (1 - \frac{1}{3})}$

Step 5.2

Apply the distributive property.

$- 4 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{8})}{3 \cdot 1 + 3 (- \frac{1}{3})}$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$- 4 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{8})}{3 \cdot 1 + 3 (\frac{- 1}{3})}$

Step 5.3.2

Cancel the common factor.

$- 4 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{8})}{3 \cdot 1 + 3 (\frac{- 1}{3})}$

Step 5.3.3

Rewrite the expression.

$- 4 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{8})}{3 \cdot 1 - 1}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $3$ by $1$ .

$- 4 \frac{3 + 3 (- {(\frac{1}{3})}^{8})}{3 \cdot 1 - 1}$

Step 5.4.2

Apply the product rule to $\frac{1}{3}$ .

$- 4 \frac{3 + 3 (- \frac{1^{8}}{3^{8}})}{3 \cdot 1 - 1}$

Step 5.4.3

Cancel the common factor of $3$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{1^{8}}{3^{8}}$ into the numerator.

$- 4 \frac{3 + 3 \frac{- 1^{8}}{3^{8}}}{3 \cdot 1 - 1}$

Step 5.4.3.2

Factor $3$ out of $3^{8}$ .

$- 4 \frac{3 + 3 \frac{- 1^{8}}{3 \cdot 3^{7}}}{3 \cdot 1 - 1}$

Step 5.4.3.3

Cancel the common factor.

$- 4 \frac{3 + 3 \frac{- 1^{8}}{3 \cdot 3^{7}}}{3 \cdot 1 - 1}$

Step 5.4.3.4

Rewrite the expression.

$- 4 \frac{3 + \frac{- 1^{8}}{3^{7}}}{3 \cdot 1 - 1}$

Step 5.4.4

One to any power is one.

$- 4 \frac{3 + \frac{- 1 \cdot 1}{3^{7}}}{3 \cdot 1 - 1}$

Step 5.4.5

Raise $3$ to the power of $7$ .

$- 4 \frac{3 + \frac{- 1 \cdot 1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.6

Multiply $- 1$ by $1$ .

$- 4 \frac{3 + \frac{- 1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.7

Move the negative in front of the fraction.

$- 4 \frac{3 - \frac{1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.8

To write $3$ as a fraction with a common denominator, multiply by $\frac{2187}{2187}$ .

$- 4 \frac{3 \cdot \frac{2187}{2187} - \frac{1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.9

Combine $3$ and $\frac{2187}{2187}$ .

$- 4 \frac{\frac{3 \cdot 2187}{2187} - \frac{1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.10

Combine the numerators over the common denominator.

$- 4 \frac{\frac{3 \cdot 2187 - 1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $3$ by $2187$ .

$- 4 \frac{\frac{6561 - 1}{2187}}{3 \cdot 1 - 1}$

Step 5.4.11.2

Subtract $1$ from $6561$ .

$- 4 \frac{\frac{6560}{2187}}{3 \cdot 1 - 1}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $3$ by $1$ .

$- 4 \frac{\frac{6560}{2187}}{3 - 1}$

Step 5.5.2

Subtract $1$ from $3$ .

$- 4 \frac{\frac{6560}{2187}}{2}$

Step 5.6

Cancel the common factor of $2$ .

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Step 5.6.1

Factor $2$ out of $- 4$ .

$2 (- 2) \frac{\frac{6560}{2187}}{2}$

Step 5.6.2

Cancel the common factor.

$2 \cdot - 2 \frac{\frac{6560}{2187}}{2}$

Step 5.6.3

Rewrite the expression.

$- 2 (\frac{6560}{2187})$

Step 5.7

Combine $- 2$ and $\frac{6560}{2187}$ .

$\frac{- 2 \cdot 6560}{2187}$

Step 5.8

Multiply $- 2$ by $6560$ .

$\frac{- 13120}{2187}$

Step 5.9

Move the negative in front of the fraction.

$- \frac{13120}{2187}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{13120}{2187}$

Decimal Form:

$- 5.99908550 \dots$

Mixed Number Form:

$- 5 \frac{2185}{2187}$