Calculus Examples

$\sum_{i = - 2}^{3} 3^{- i}$

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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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Multiply by $1$ .

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

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

Step 2.2.1.2.4

Divide $3^{- (i + 1) + i}$ by $1$ .

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Apply the distributive property.

$r = 3^{- i - 1 \cdot 1 + i}$

Step 2.2.2.2

Multiply $- 1$ by $1$ .

$r = 3^{- i - 1 + i}$

Step 2.2.3

Combine the opposite terms in $- i - 1 + i$ .

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Step 2.2.3.1

Add $- i$ and $i$ .

$r = 3^{0 - 1}$

Step 2.2.3.2

Subtract $1$ from $0$ .

$r = 3^{- 1}$

Step 2.2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

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 $- 2$ for $i$ into $3^{- i}$ .

$a = 3^{2}$

Step 3.2

Raise $3$ to the power of $2$ .

$a = 9$

Step 4

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

$9 \frac{1 - {(\frac{1}{3})}^{6}}{1 - \frac{1}{3}}$

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 $\frac{1 - {(\frac{1}{3})}^{6}}{1 - \frac{1}{3}}$ by $\frac{3}{3}$ .

$9 (\frac{3}{3} \cdot \frac{1 - {(\frac{1}{3})}^{6}}{1 - \frac{1}{3}})$

Step 5.1.2

Combine.

$9 \frac{3 (1 - {(\frac{1}{3})}^{6})}{3 (1 - \frac{1}{3})}$

Step 5.2

Apply the distributive property.

$9 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{6})}{3 \cdot 1 + 3 (- \frac{1}{3})}$

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.

$9 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{6})}{3 \cdot 1 + 3 (\frac{- 1}{3})}$

Step 5.3.2

Cancel the common factor.

$9 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{6})}{3 \cdot 1 + 3 (\frac{- 1}{3})}$

Step 5.3.3

Rewrite the expression.

$9 \frac{3 \cdot 1 + 3 (- {(\frac{1}{3})}^{6})}{3 \cdot 1 - 1}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $3$ by $1$ .

$9 \frac{3 + 3 (- {(\frac{1}{3})}^{6})}{3 \cdot 1 - 1}$

Step 5.4.2

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

$9 \frac{3 + 3 (- \frac{1^{6}}{3^{6}})}{3 \cdot 1 - 1}$

Step 5.4.3

Cancel the common factor of $3$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{1^{6}}{3^{6}}$ into the numerator.

$9 \frac{3 + 3 \frac{- 1^{6}}{3^{6}}}{3 \cdot 1 - 1}$

Step 5.4.3.2

Factor $3$ out of $3^{6}$ .

$9 \frac{3 + 3 \frac{- 1^{6}}{3 \cdot 3^{5}}}{3 \cdot 1 - 1}$

Step 5.4.3.3

Cancel the common factor.

$9 \frac{3 + 3 \frac{- 1^{6}}{3 \cdot 3^{5}}}{3 \cdot 1 - 1}$

Step 5.4.3.4

Rewrite the expression.

$9 \frac{3 + \frac{- 1^{6}}{3^{5}}}{3 \cdot 1 - 1}$

Step 5.4.4

One to any power is one.

$9 \frac{3 + \frac{- 1 \cdot 1}{3^{5}}}{3 \cdot 1 - 1}$

Step 5.4.5

Raise $3$ to the power of $5$ .

$9 \frac{3 + \frac{- 1 \cdot 1}{243}}{3 \cdot 1 - 1}$

Step 5.4.6

Multiply $- 1$ by $1$ .

$9 \frac{3 + \frac{- 1}{243}}{3 \cdot 1 - 1}$

Step 5.4.7

Move the negative in front of the fraction.

$9 \frac{3 - \frac{1}{243}}{3 \cdot 1 - 1}$

Step 5.4.8

To write $3$ as a fraction with a common denominator, multiply by $\frac{243}{243}$ .

$9 \frac{3 \cdot \frac{243}{243} - \frac{1}{243}}{3 \cdot 1 - 1}$

Step 5.4.9

Combine $3$ and $\frac{243}{243}$ .

$9 \frac{\frac{3 \cdot 243}{243} - \frac{1}{243}}{3 \cdot 1 - 1}$

Step 5.4.10

Combine the numerators over the common denominator.

$9 \frac{\frac{3 \cdot 243 - 1}{243}}{3 \cdot 1 - 1}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $3$ by $243$ .

$9 \frac{\frac{729 - 1}{243}}{3 \cdot 1 - 1}$

Step 5.4.11.2

Subtract $1$ from $729$ .

$9 \frac{\frac{728}{243}}{3 \cdot 1 - 1}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $3$ by $1$ .

$9 \frac{\frac{728}{243}}{3 - 1}$

Step 5.5.2

Subtract $1$ from $3$ .

$9 \frac{\frac{728}{243}}{2}$

Step 5.6

Multiply the numerator by the reciprocal of the denominator.

$9 (\frac{728}{243} \cdot \frac{1}{2})$

Step 5.7

Cancel the common factor of $2$ .

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Step 5.7.1

Factor $2$ out of $728$ .

$9 (\frac{2 (364)}{243} \cdot \frac{1}{2})$

Step 5.7.2

Cancel the common factor.

$9 (\frac{2 \cdot 364}{243} \cdot \frac{1}{2})$

Step 5.7.3

Rewrite the expression.

$9 (\frac{364}{243})$

Step 5.8

Cancel the common factor of $9$ .

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Step 5.8.1

Factor $9$ out of $243$ .

$9 \frac{364}{9 (27)}$

Step 5.8.2

Cancel the common factor.

$9 \frac{364}{9 \cdot 27}$

Step 5.8.3

Rewrite the expression.

$\frac{364}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{364}{27}$

Decimal Form:

$13.\overline{481}$

Mixed Number Form:

$13 \frac{13}{27}$