Calculus Examples

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

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 $r$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.2.4

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

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

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 ${(\frac{1}{3})}^{n}$ .

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

Step 3.2

Simplify.

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

Step 4

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

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

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 $\frac{3}{3}$ .

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

Step 5.1.2

Combine.

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

Step 5.2

Apply the distributive property.

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

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.

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

Step 5.3.2

Cancel the common factor.

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

Step 5.3.3

Rewrite the expression.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $3$ by $1$ .

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

Step 5.4.2

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

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

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.

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

Step 5.4.3.2

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

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

Step 5.4.3.3

Cancel the common factor.

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

Step 5.4.3.4

Rewrite the expression.

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

Step 5.4.4

One to any power is one.

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

Step 5.4.5

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

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

Step 5.4.6

Multiply $- 1$ by $1$ .

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

Step 5.4.7

Move the negative in front of the fraction.

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

Step 5.4.8

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

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

Step 5.4.9

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

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

Step 5.4.10

Combine the numerators over the common denominator.

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

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $3$ by $2187$ .

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

Step 5.4.11.2

Subtract $1$ from $6561$ .

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

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $3$ by $1$ .

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

Step 5.5.2

Subtract $1$ from $3$ .

$\frac{1}{3} \cdot \frac{\frac{6560}{2187}}{2}$

Step 5.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} (\frac{6560}{2187} \cdot \frac{1}{2})$

Step 5.7

Cancel the common factor of $2$ .

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Step 5.7.1

Factor $2$ out of $6560$ .

$\frac{1}{3} (\frac{2 (3280)}{2187} \cdot \frac{1}{2})$

Step 5.7.2

Cancel the common factor.

$\frac{1}{3} (\frac{2 \cdot 3280}{2187} \cdot \frac{1}{2})$

Step 5.7.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{3280}{2187}$

Step 5.8

Multiply $\frac{1}{3} \cdot \frac{3280}{2187}$ .

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Step 5.8.1

Multiply $\frac{1}{3}$ by $\frac{3280}{2187}$ .

$\frac{3280}{3 \cdot 2187}$

Step 5.8.2

Multiply $3$ by $2187$ .

$\frac{3280}{6561}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3280}{6561}$

Decimal Form:

$0.49992379 \dots$